{"id":65100,"date":"2025-08-18T09:45:51","date_gmt":"2025-08-18T06:45:51","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=65100"},"modified":"2026-10-23T17:00:30","modified_gmt":"2026-10-23T14:00:30","slug":"binomial-coefficient","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/binomial-coefficient\/","title":{"rendered":"n Choose k Calculator \u2013 Binomial Coefficient"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Calculate \u201cn Choose k\u201d Combinations<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"65100\" data-favorite-title=\"Binomial Coefficient\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/binomial-coefficient\/\" data-favorite-excerpt=\"Calculate the binomial coefficient C(n, k) \u2014 the number of ways to choose k items from n without regard to order. Enter n and k and get an instant result with the formula and a worked example.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-algebra\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/mathematics\/binomial-coefficient\/\" method=\"POST\" class=\"calculator\" id=\"calculator-65100\" data-post=\"65100\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">n (total items)<\/label><div class=\"term\">n<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"n_total_items\" id=\"field-1\" aria-describedby=\"calc-info-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"5\"\/><small id=\"calc-info-1\">Enter the total number of items (n)<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">k (selected items)<\/label><div class=\"term\">k<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"k_selected_items\" id=\"field-2\" aria-describedby=\"calc-info-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"2\"\/><small id=\"calc-info-2\">Enter the number of items to choose (k)<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"65100\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Your Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Binomial Coefficient<\/span><span class=\"result-term\">C(n, k)<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copy link to this calculation<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"65100\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">1<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">1<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Binomial Coefficient Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ \\binom{n}{k} = \\frac{n!}{k!\\,(n-k)!} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(n\\) \u2014 total number of items (a non-negative integer)<\/li>\n<li>\\(k\\) \u2014 number of items to choose (\\(0 \\le k \\le n\\))<\/li>\n<li>\\(n!\\) \u2014 factorial of \\(n\\), the product \\(1 \\times 2 \\times \\dots \\times n\\), with \\(0! = 1\\)<\/li>\n<\/ul>\n<p>The formula counts how many distinct subsets of size \\(k\\) can be formed from \\(n\\) items when order does not matter. It is the same coefficient that appears in Pascal's Triangle and in the expansion of \\((a+b)^n\\).<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Binomial Coefficient \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a committee of 2 people must be chosen from a group of 5. How many different committees are possible? Here \\(n = 5\\) and \\(k = 2\\).<\/p>\n<p>\\[ \\binom{5}{2} = \\frac{5!}{2!\\,(5-2)!} = \\frac{120}{2 \\times 6} = \\frac{120}{12} = 10 \\]<\/p>\n<p><strong>Result:<\/strong> \\(\\binom{5}{2} = 10\\) \u2014 there are 10 ways to choose 2 people out of 5.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Enter the total number of items in the field <strong>n (total items)<\/strong>.<\/li>\n<li>Enter the number of items to choose in the field <strong>k (selected items)<\/strong>. The value of \\(k\\) cannot exceed \\(n\\).<\/li>\n<li>Read the result under <strong>Binomial Coefficient<\/strong> \u2014 this is \\(\\binom{n}{k}\\), the number of possible selections.<\/li>\n<\/ol>\n<h2>What Is the Binomial Coefficient?<\/h2>\n<p>The binomial coefficient \\(\\binom{n}{k}\\), read aloud as \"n choose k\", answers a fundamental counting question: in how many ways can you pick \\(k\\) objects from a set of \\(n\\) objects when the order of the picked objects does not matter? If you have 5 books and want to take 2 on a trip, it does not matter which one you grab first \u2014 what matters is which pair ends up in your bag. That is why binomial coefficients count <em>combinations<\/em>, as opposed to <em>permutations<\/em>, where order matters and the count would be larger.<\/p>\n<p>The name comes from the binomial theorem: the coefficients in the expansion of \\((a+b)^n\\) are exactly \\(\\binom{n}{0}, \\binom{n}{1}, \\dots, \\binom{n}{n}\\). For example, \\((a+b)^2 = a^2 + 2ab + b^2\\), where the coefficient 2 equals \\(\\binom{2}{1}\\). If you want to expand products of two-term expressions, the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/multiplying-binomials\/\">Multiplying Binomials calculator<\/a> and the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/foil-method\/\">FOIL method calculator<\/a> handle that step directly.<\/p>\n<p>Binomial coefficients are also the building blocks of probability. If an experiment has \\(n\\) independent trials, each succeeding with probability \\(p\\), the chance of exactly \\(k\\) successes is \\(\\binom{n}{k} p^k (1-p)^{n-k}\\). The coefficient counts which of the \\(n\\) trials are the successful ones, which is why it appears whenever outcomes are counted in groups.<\/p>\n<p>Each row of Pascal's Triangle lists the coefficients for one value of \\(n\\): row 5 is 1, 5, 10, 10, 5, 1, which contains \\(\\binom{5}{2} = 10\\). The triangle makes the symmetry \\(\\binom{n}{k} = \\binom{n}{n-k}\\) and Pascal's rule, \\(\\binom{n}{k} = \\binom{n-1}{k-1} + \\binom{n-1}{k}\\), easy to see. The same triangle also connects to the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/fibonacci-series\/\">Fibonacci series<\/a>, since the sums of its shallow diagonals produce the Fibonacci numbers.<\/p>\n<h2>Interpreting the Result<\/h2>\n<p>The result is always a whole number, because you cannot have a fraction of a selection. Two special cases are worth remembering: \\(\\binom{n}{0} = 1\\) (there is exactly one way to choose nothing) and \\(\\binom{n}{n} = 1\\) (one way to choose everything). If \\(k &gt; n\\), the selection is impossible and the calculator shows an error rather than a number.<\/p>\n<p>Useful identities to check your work: \\(\\binom{n}{1} = n\\), \\(\\binom{n}{2} = \\frac{n(n-1)}{2}\\), and by symmetry \\(\\binom{n}{k} = \\binom{n}{n-k}\\), so \\(\\binom{5}{3} = \\binom{5}{2} = 10\\).<\/p>\n<h2>Assumptions and Limitations<\/h2>\n<ul>\n<li><strong>Integers only.<\/strong> Both \\(n\\) and \\(k\\) must be non-negative integers. The combinatorial meaning of \"choosing items\" does not extend to fractional inputs.<\/li>\n<li><strong>\\(k \\le n\\).<\/strong> If the number of items to choose exceeds the number available, the calculator reports an error; mathematically, \\(\\binom{n}{k} = 0\\) for \\(k &gt; n\\).<\/li>\n<li><strong>Order is ignored.<\/strong> This calculator counts unordered selections. If the order of the chosen items matters, you need permutations, \\(P(n,k) = \\frac{n!}{(n-k)!}\\), which is always greater than or equal to \\(\\binom{n}{k}\\).<\/li>\n<li><strong>No repetition.<\/strong> Each of the \\(n\\) items can be chosen at most once. Selections with repetition allowed use a different formula, \\(\\binom{n+k-1}{k}\\).<\/li>\n<li><strong>Size limits.<\/strong> Factorials grow extremely fast; \\(70!\\) already exceeds \\(10^{100}\\). For very large \\(n\\), results may be limited by number precision even though the mathematical value is exact.<\/li>\n<\/ul>\n<h2>Common Values Reference<\/h2>\n<table>\n<thead>\n<tr>\n<th>\\(n\\)<\/th>\n<th>\\(k\\)<\/th>\n<th>\\(\\binom{n}{k}\\)<\/th>\n<th>Interpretation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>4<\/td>\n<td>2<\/td>\n<td>6<\/td>\n<td>Pairs from 4 items<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>2<\/td>\n<td>10<\/td>\n<td>Pairs from 5 items<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>3<\/td>\n<td>20<\/td>\n<td>Triples from 6 items<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>5<\/td>\n<td>252<\/td>\n<td>Half of 10 items<\/td>\n<\/tr>\n<tr>\n<td>52<\/td>\n<td>5<\/td>\n<td>2,598,960<\/td>\n<td>5-card poker hands<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The last row shows why binomial coefficients are the standard tool for card problems: the number of distinct 5-card hands from a 52-card deck is exactly \\(\\binom{52}{5}\\).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Calculate the binomial coefficient C(n, k) \u2014 the number of ways to choose k items from n without regard to order. Enter n and k and get an instant result with the formula and a worked example.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[11],"class_list":["post-65100","post","type-post","status-publish","format-standard","hentry","category-mathematics","tag-algebra"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65100","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=65100"}],"version-history":[{"count":4,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65100\/revisions"}],"predecessor-version":[{"id":67499,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65100\/revisions\/67499"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=65100"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=65100"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=65100"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}