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Tổ hợp liệt kê – Wikipedia tiếng Việt

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user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%C4%90%E1%BA%B7c_bi%E1%BB%87t:M%E1%BB%9F_t%C3%A0i_kho%E1%BA%A3n&amp;returnto=T%E1%BB%95+h%E1%BB%A3p+li%E1%BB%87t+k%C3%AA" title="Bạn được khuyến khích mở tài khoản và đăng nhập; tuy nhiên, không bắt buộc phải có tài khoản" class=""><span>Tạo tài khoản</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%C4%90%E1%BA%B7c_bi%E1%BB%87t:%C4%90%C4%83ng_nh%E1%BA%ADp&amp;returnto=T%E1%BB%95+h%E1%BB%A3p+li%E1%BB%87t+k%C3%AA" title="Đăng nhập sẽ có lợi hơn, tuy nhiên không bắt buộc. 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[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Đăng nhập</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Trang dành cho người dùng chưa đăng nhập <a href="/wiki/Tr%E1%BB%A3_gi%C3%BAp:Gi%E1%BB%9Bi_thi%E1%BB%87u" aria-label="Tìm hiểu thêm về sửa đổi"><span>tìm hiểu thêm</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:%C4%90%C3%B3ng_g%C3%B3p_c%E1%BB%A7a_t%C3%B4i" title="Danh sách các sửa đổi được thực hiện qua địa chỉ IP này [y]" accesskey="y"><span>Đóng góp</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Th%E1%BA%A3o_lu%E1%BA%ADn_t%C3%B4i" title="Thảo luận với địa chỉ IP này [n]" accesskey="n"><span>Thảo luận cho địa chỉ IP này</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"><!-- CentralNotice --></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Trang Web"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Nội dung" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Nội dung</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">chuyển sang thanh bên</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ẩn</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Đầu</div> </a> </li> <li id="toc-Xem_thêm" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Xem_thêm"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Xem thêm</span> </div> </a> <ul id="toc-Xem_thêm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tham_khảo" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tham_khảo"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Tham khảo</span> </div> </a> <ul id="toc-Tham_khảo-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Nội dung" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Đóng mở mục lục" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Đóng mở mục lục</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Tổ hợp liệt kê</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Xem bài viết trong ngôn ngữ khác. Bài có sẵn trong 11 ngôn ngữ" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-11" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">11 ngôn ngữ</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%B1%D9%83%D8%A8%D9%8A%D8%A7%D8%AA_%D8%AA%D8%B9%D8%AF%D8%A7%D8%AF%D9%8A%D8%A9" title="تركبيات تعدادية – Tiếng Ả Rập" lang="ar" hreflang="ar" data-title="تركبيات تعدادية" data-language-autonym="العربية" data-language-local-name="Tiếng Ả Rập" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Abz%C3%A4hlende_Kombinatorik" title="Abzählende Kombinatorik – Tiếng Đức" lang="de" hreflang="de" data-title="Abzählende Kombinatorik" data-language-autonym="Deutsch" data-language-local-name="Tiếng Đức" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Enumerative_combinatorics" title="Enumerative combinatorics – Tiếng Anh" lang="en" hreflang="en" data-title="Enumerative combinatorics" data-language-autonym="English" data-language-local-name="Tiếng Anh" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Combinatoria_enumerativa" title="Combinatoria enumerativa – Tiếng Tây Ban Nha" lang="es" hreflang="es" data-title="Combinatoria enumerativa" data-language-autonym="Español" data-language-local-name="Tiếng Tây Ban Nha" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%B1%DA%A9%DB%8C%D8%A8%DB%8C%D8%A7%D8%AA_%D8%B4%D9%85%D8%A7%D8%B1%D8%B4%DB%8C" title="ترکیبیات شمارشی – Tiếng Ba Tư" lang="fa" hreflang="fa" data-title="ترکیبیات شمارشی" data-language-autonym="فارسی" data-language-local-name="Tiếng Ba Tư" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Elemi_kombinatorika" title="Elemi kombinatorika – Tiếng Hungary" lang="hu" hreflang="hu" data-title="Elemi kombinatorika" data-language-autonym="Magyar" data-language-local-name="Tiếng Hungary" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%95%B0%E3%81%88%E4%B8%8A%E3%81%92%E6%95%B0%E5%AD%A6" title="数え上げ数学 – Tiếng Nhật" lang="ja" hreflang="ja" data-title="数え上げ数学" data-language-autonym="日本語" data-language-local-name="Tiếng Nhật" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Combinat%C3%B3ria_enumerativa" title="Combinatória enumerativa – Tiếng Bồ Đào Nha" lang="pt" hreflang="pt" data-title="Combinatória enumerativa" data-language-autonym="Português" data-language-local-name="Tiếng Bồ Đào Nha" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%B5%D1%80%D0%B5%D1%87%D0%B8%D1%81%D0%BB%D0%B8%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%BA%D0%BE%D0%BC%D0%B1%D0%B8%D0%BD%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BA%D0%B0" title="Перечислительная комбинаторика – Tiếng Nga" lang="ru" hreflang="ru" data-title="Перечислительная комбинаторика" data-language-autonym="Русский" data-language-local-name="Tiếng Nga" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D1%83%D0%BC%D0%B5%D1%80%D0%B0%D1%86%D1%96%D0%B9%D0%BD%D0%B0_%D0%BA%D0%BE%D0%BC%D0%B1%D1%96%D0%BD%D0%B0%D1%82%D0%BE%D1%80%D0%B8%D0%BA%D0%B0" title="Нумераційна комбінаторика – Tiếng Ukraina" lang="uk" hreflang="uk" data-title="Нумераційна комбінаторика" data-language-autonym="Українська" data-language-local-name="Tiếng Ukraina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%BB%84%E5%90%88%E8%AE%A1%E6%95%B0" title="组合计数 – Tiếng Trung" lang="zh" hreflang="zh" data-title="组合计数" data-language-autonym="中文" data-language-local-name="Tiếng Trung" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q336787#sitelinks-wikipedia" title="Sửa liên kết giữa ngôn ngữ" class="wbc-editpage">Sửa liên kết</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Không gian tên"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/T%E1%BB%95_h%E1%BB%A3p_li%E1%BB%87t_k%C3%AA" title="Xem bài viết [c]" accesskey="c"><span>Bài viết</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Th%E1%BA%A3o_lu%E1%BA%ADn:T%E1%BB%95_h%E1%BB%A3p_li%E1%BB%87t_k%C3%AA&amp;action=edit&amp;redlink=1" rel="discussion" class="new" title="Thảo luận về trang này (trang không tồn tại) [t]" accesskey="t"><span>Thảo luận</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Thay đổi biến thể ngôn ngữ" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tiếng Việt</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Giao diện"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/T%E1%BB%95_h%E1%BB%A3p_li%E1%BB%87t_k%C3%AA"><span>Đọc</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=T%E1%BB%95_h%E1%BB%A3p_li%E1%BB%87t_k%C3%AA&amp;veaction=edit" title="Sửa đổi trang này [v]" accesskey="v"><span>Sửa đổi</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=T%E1%BB%95_h%E1%BB%A3p_li%E1%BB%87t_k%C3%AA&amp;action=edit" title="Sửa đổi mã nguồn của trang này [e]" accesskey="e"><span>Sửa mã nguồn</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=T%E1%BB%95_h%E1%BB%A3p_li%E1%BB%87t_k%C3%AA&amp;action=history" title="Các phiên bản cũ của trang này [h]" accesskey="h"><span>Xem lịch sử</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Công cụ trang"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Công cụ" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Công cụ</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Công cụ</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">chuyển sang thanh bên</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">ẩn</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Thêm tùy chọn" > <div class="vector-menu-heading"> Tác vụ </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/T%E1%BB%95_h%E1%BB%A3p_li%E1%BB%87t_k%C3%AA"><span>Đọc</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=T%E1%BB%95_h%E1%BB%A3p_li%E1%BB%87t_k%C3%AA&amp;veaction=edit" title="Sửa đổi trang này [v]" accesskey="v"><span>Sửa đổi</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=T%E1%BB%95_h%E1%BB%A3p_li%E1%BB%87t_k%C3%AA&amp;action=edit" title="Sửa đổi mã nguồn của trang này [e]" accesskey="e"><span>Sửa mã nguồn</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=T%E1%BB%95_h%E1%BB%A3p_li%E1%BB%87t_k%C3%AA&amp;action=history"><span>Xem lịch sử</span></a></li> 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PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=T%E1%BB%95_h%E1%BB%A3p_li%E1%BB%87t_k%C3%AA&amp;printable=yes" title="Bản để in ra của trang [p]" accesskey="p"><span>Bản để in ra</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Tại dự án khác </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q336787" title="Liên kết đến khoản mục kết nối trong kho dữ liệu [g]" accesskey="g"><span>Khoản mục Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Công cụ trang"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Giao diện"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Giao diện</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">chuyển sang thanh bên</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ẩn</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Bách khoa toàn thư mở Wikipedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="vi" dir="ltr"><p><b>Tổ hợp liệt kê</b> là một phần của <a href="/wiki/To%C3%A1n_h%E1%BB%8Dc_t%E1%BB%95_h%E1%BB%A3p" title="Toán học tổ hợp">toán học tổ hợp</a> nghiên cứu số cách mà các mẫu nhất định có thể được hình thành. Hai ví dụ về loại vấn đề này là đếm số <a href="/wiki/T%E1%BB%95_h%E1%BB%A3p_(to%C3%A1n_h%E1%BB%8Dc)" title="Tổ hợp (toán học)">tổ hợp</a> và đếm số <a href="/wiki/Ho%C3%A1n_v%E1%BB%8B" title="Hoán vị">hoán vị</a>. Nói chung, cho một tập hợp vô hạn các tập hữu hạn <i>S</i><sub><i>i</i></sub> được lập chỉ mục bằng <a href="/wiki/S%E1%BB%91_t%E1%BB%B1_nhi%C3%AAn" title="Số tự nhiên">các số tự nhiên</a>, tổ hợp liệt kê tìm cách mô tả một hàm đếm số đối tượng trong <i>S</i><sub><i>n</i></sub> cho từng số <i>n</i>. Mặc dù đếm số lượng các phần tử trong một tập hợp là <a href="/wiki/To%C3%A1n_%C4%91%E1%BB%91" title="Toán đố">bài toán</a> khá rộng, nhiều vấn đề phát sinh trong các ứng dụng có một mô tả <a href="/wiki/To%C3%A1n_h%E1%BB%8Dc_t%E1%BB%95_h%E1%BB%A3p" title="Toán học tổ hợp">tổ hợp</a> tương đối đơn giản. Phương pháp 12 cách cung cấp một khuôn khổ thống nhất để đếm các <a href="/wiki/Ho%C3%A1n_v%E1%BB%8B" title="Hoán vị">hoán vị</a>, <a href="/wiki/T%E1%BB%95_h%E1%BB%A3p_(to%C3%A1n_h%E1%BB%8Dc)" title="Tổ hợp (toán học)">tổ hợp</a> và phân vùng. </p><p>Những hàm như vậy đơn giản nhất đều là các công thức đóng, nghĩa là có thể diễn đạt bằng một hỗn hợp các hàm sơ cấp như <a href="/wiki/Giai_th%E1%BB%ABa" title="Giai thừa">giai thừa</a>, hàm mũ v.v... Chẳng hạn, như chỉ ra dưới đây, số lượng các cách sắp xếp của <i>n</i> lá bài là <i>f</i>(<i>n</i>) = <i>n</i>!. Bài toán tìm kiếm một công thức đóng được gọi là liệt kê đại số, và nó thường xuyên bao gồm việc tính toán kiểu đệ quy hoặc hàm chuỗi và dùng nó để đạt tới dạng hàm mong muốn. </p><p>Thông thường, một công thức khép kín phức tạp mang lại ít hiểu biết sâu sắc về hành vi của việc đếm khi số lượng vật thể đếm tăng lên. Trong những trường hợp này, một phép xấp xỉ tiệm cận đơn giản có thể thích hợp hơn. Một hàm <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4ad18070e494503403daf39398e711c1378348e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.32ex; height:2.843ex;" alt="{\displaystyle g(n)}"></span> Là một xấp xỉ tiệm cận của <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c49fad1eccc4e9af1e4f23f32efdc3ac4da973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\displaystyle f(n)}"></span> nếu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)/g(n)\rightarrow 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">&#x2192;<!-- → --></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)/g(n)\rightarrow 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc5c8cce354147041272734e91375a92287508ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.742ex; height:2.843ex;" alt="{\displaystyle f(n)/g(n)\rightarrow 1}"></span> khi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9702f04f2d0e5b887b99faeeffb0c4cfd8263eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\rightarrow \infty }"></span>. Trong trường hợp này chúng ta viết <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)\sim g(n).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>&#x223C;<!-- ∼ --></mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)\sim g(n).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65047cadb135e6ada9fbeaf9c4d5603189b39e8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.935ex; height:2.843ex;" alt="{\displaystyle f(n)\sim g(n).\,}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Xem_thêm"><span id="Xem_th.C3.AAm"></span>Xem thêm</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T%E1%BB%95_h%E1%BB%A3p_li%E1%BB%87t_k%C3%AA&amp;veaction=edit&amp;section=1" title="Sửa đổi phần “Xem thêm”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=T%E1%BB%95_h%E1%BB%A3p_li%E1%BB%87t_k%C3%AA&amp;action=edit&amp;section=1" title="Sửa mã nguồn tại đề mục: Xem thêm"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/B%E1%BB%95_%C4%91%E1%BB%81_Burnside" title="Bổ đề Burnside">Bổ đề Burnside</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Tham_khảo"><span id="Tham_kh.E1.BA.A3o"></span>Tham khảo</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T%E1%BB%95_h%E1%BB%A3p_li%E1%BB%87t_k%C3%AA&amp;veaction=edit&amp;section=2" title="Sửa đổi phần “Tham khảo”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=T%E1%BB%95_h%E1%BB%A3p_li%E1%BB%87t_k%C3%AA&amp;action=edit&amp;section=2" title="Sửa mã nguồn tại đề mục: Tham khảo"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r71728118">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist" style="list-style-type: decimal;"> </div> <ul><li><a href="/w/index.php?title=Doron_Zeilberger&amp;action=edit&amp;redlink=1" class="new" title="Doron Zeilberger (trang không tồn tại)">Zeilberger, D.</a>, <a rel="nofollow" class="external text" href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/enuPCM.pdf">Enumerative and Algebraic Combinatorics</a></li> <li>Bjorner, A. and <a href="/w/index.php?title=Richard_P._Stanley&amp;action=edit&amp;redlink=1" class="new" title="Richard P. Stanley (trang không tồn tại)">Stanley, R. P.</a>, <a rel="nofollow" class="external text" href="http://www-math.mit.edu/~rstan/papers/comb.pdf"><i>A Combinatorial Miscellany</i></a></li> <li>Graham, R.L., <a href="/w/index.php?title=Martin_Gr%C3%B6tschel&amp;action=edit&amp;redlink=1" class="new" title="Martin Grötschel (trang không tồn tại)">Grötschel M.</a>, and Lovász L., eds. (1996). <i>Handbook of Combinatorics</i>, Volumes 1 and 2. Elsevier (North-Holland), Amsterdam, and MIT Press, Cambridge, Mass. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Ngu%E1%BB%93n_s%C3%A1ch/0-262-07169-X" title="Đặc biệt:Nguồn sách/0-262-07169-X">0-262-07169-X</a>.</li> <li><style data-mw-deduplicate="TemplateStyles:r67233549">.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"“""”""‘""’"}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFJoseph1994" class="citation book cs1">Joseph, George Gheverghese (1994) [1991]. <i>The Crest of the Peacock: Non-European Roots of Mathematics</i> (ấn bản thứ 2). London: <a href="/wiki/Penguin_Books" title="Penguin Books">Penguin Books</a>. <a href="/wiki/ISBN" title="ISBN">ISBN</a>&#160;<a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Ngu%E1%BB%93n_s%C3%A1ch/0-14-012529-9" title="Đặc biệt:Nguồn sách/0-14-012529-9"><bdi>0-14-012529-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=The+Crest+of+the+Peacock%3A+Non-European+Roots+of+Mathematics&amp;rft.place=London&amp;rft.edition=2nd&amp;rft.pub=Penguin+Books&amp;rft.date=1994&amp;rft.isbn=0-14-012529-9&amp;rft.aulast=Joseph&amp;rft.aufirst=George+Gheverghese&amp;rfr_id=info%3Asid%2Fvi.wikipedia.org%3AT%E1%BB%95+h%E1%BB%A3p+li%E1%BB%87t+k%C3%AA" class="Z3988"></span></li> <li>Loehr, Nicholas A. (2011). <a rel="nofollow" class="external text" href="http://www.math.vt.edu/people/nloehr/bijbook.html">Bijective Combinatorics</a> <a rel="nofollow" class="external text" href="https://wayback.archive-it.org/all/20151023194824/http://www.math.vt.edu/people/nloehr/bijbook.html">Lưu trữ</a> <span style="font-size:100%" class="error citation-comment">[Date missing]</span> tại <a href="/wiki/Archive-It" class="mw-redirect" title="Archive-It">Archive-It</a>. <a rel="nofollow" class="external text" href="http://www.crcpress.com">CRC Press</a>. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Ngu%E1%BB%93n_s%C3%A1ch/143984884X" title="Đặc biệt:Nguồn sách/143984884X">143984884X</a>, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Ngu%E1%BB%93n_s%C3%A1ch/978-1439848845" title="Đặc biệt:Nguồn sách/978-1439848845">978-1439848845</a>.</li> <li><a href="/w/index.php?title=Richard_P._Stanley&amp;action=edit&amp;redlink=1" class="new" title="Richard P. Stanley (trang không tồn tại)">Stanley, Richard P.</a> (1997, 1999). <a rel="nofollow" class="external text" href="http://www-math.mit.edu/~rstan/ec/"><i>Enumerative Combinatorics</i>, Volumes 1 and 2</a>. <a href="/wiki/Cambridge_University_Press" class="mw-redirect" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Ngu%E1%BB%93n_s%C3%A1ch/0-521-55309-1" title="Đặc biệt:Nguồn sách/0-521-55309-1">0-521-55309-1</a>, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Ngu%E1%BB%93n_s%C3%A1ch/0-521-56069-1" title="Đặc biệt:Nguồn sách/0-521-56069-1">0-521-56069-1</a>.</li> <li><a rel="nofollow" class="external text" href="http://encyclopedia.jrank.org/CLI_COM/COMBINATORIAL_ANALYSIS.html">Combinatorial Analysis</a> – an article in <a href="/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">Encyclopædia Britannica Eleventh Edition</a></li> <li><a href="/w/index.php?title=John_Riordan_(mathematician)&amp;action=edit&amp;redlink=1" class="new" title="John Riordan (mathematician) (trang không tồn tại)">Riordan, John</a> (1958). <i>An Introduction to Combinatorial Analysis</i>, Wiley &amp; Sons, New York (republished).</li> <li>Riordan, John (1968). <i>Combinatorial identities</i>, Wiley &amp; Sons, New York (republished).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFWilf1994" class="citation book cs1"><a href="/w/index.php?title=Herbert_Wilf&amp;action=edit&amp;redlink=1" class="new" title="Herbert Wilf (trang không tồn tại)">Wilf, Herbert S.</a> (1994). <a rel="nofollow" class="external text" href="http://www.math.upenn.edu/%7Ewilf/DownldGF.html"><i>Generatingfunctionology</i></a> (ấn bản thứ 2). Boston, MA: Academic Press. <a href="/wiki/ISBN" title="ISBN">ISBN</a>&#160;<a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Ngu%E1%BB%93n_s%C3%A1ch/0-12-751956-4" title="Đặc biệt:Nguồn sách/0-12-751956-4"><bdi>0-12-751956-4</bdi></a>. <a href="/w/index.php?title=Zbl_(%C4%91%E1%BB%8Bnh_danh)&amp;action=edit&amp;redlink=1" class="new" title="Zbl (định danh) (trang không tồn tại)">Zbl</a>&#160;<a rel="nofollow" class="external text" href="//zbmath.org/?format=complete&amp;q=an:0831.05001">0831.05001</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Generatingfunctionology&amp;rft.place=Boston%2C+MA&amp;rft.edition=2nd&amp;rft.pub=Academic+Press&amp;rft.date=1994&amp;rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0831.05001%23id-name%3DZbl&amp;rft.isbn=0-12-751956-4&amp;rft.aulast=Wilf&amp;rft.aufirst=Herbert+S.&amp;rft_id=http%3A%2F%2Fwww.math.upenn.edu%2F%257Ewilf%2FDownldGF.html&amp;rfr_id=info%3Asid%2Fvi.wikipedia.org%3AT%E1%BB%95+h%E1%BB%A3p+li%E1%BB%87t+k%C3%AA" class="Z3988"></span></li></ul> <!-- NewPP limit report Parsed by mw‐web.codfw.main‐cc877b49b‐b5csr Cached time: 20241127121810 Cache expiry: 2592000 Reduced expiry: false Complications: [vary‐revision‐sha1] CPU time usage: 0.126 seconds Real time usage: 0.198 seconds Preprocessor visited node count: 279/1000000 Post‐expand include size: 5059/2097152 bytes Template argument size: 374/2097152 bytes Highest expansion depth: 8/100 Expensive parser function count: 0/500 Unstrip recursion depth: 0/20 Unstrip post‐expand size: 5125/5000000 bytes Lua time usage: 0.067/10.000 seconds Lua memory usage: 2469626/52428800 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 128.046 1 -total 61.12% 78.261 2 Bản_mẫu:Chú_thích_sách 18.13% 23.214 5 Bản_mẫu:ISBN 10.09% 12.916 1 Bản_mẫu:Webarchive 10.08% 12.910 1 Bản_mẫu:Tham_khảo 2.03% 2.593 5 Bản_mẫu:Main_other --> <!-- Saved in parser cache with key viwiki:pcache:7508996:|#|:idhash:canonical and timestamp 20241127121810 and revision id 66878194. 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