{"id":61475,"date":"2021-02-02T16:22:03","date_gmt":"2021-02-02T14:22:03","guid":{"rendered":"https:\/\/wpcalc.com\/?p=61475"},"modified":"2026-09-30T20:49:12","modified_gmt":"2026-09-30T17:49:12","slug":"stirlings-factorial","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/stirlings-factorial\/","title":{"rendered":"Stirling\u2019s Factorial Approximation"},"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\">Stirling&#039;s Approximation Calculator<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"61475\" data-favorite-title=\"Stirling\u2019s Factorial Approximation\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/stirlings-factorial\/\" data-favorite-excerpt=\"Estimate the factorial of a large number instantly with Stirling&#039;s approximation: enter n and get n! \u2248 \u221a(2\u03c0n)(n\/e)\u207f in scientific notation. Ideal for statistics, probability, and combinatorics.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-arithmetic\">\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\/stirlings-factorial\/\" method=\"POST\" class=\"calculator\" id=\"calculator-61475\" data-post=\"61475\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">Number of elements (n)<\/label><input type=\"text\" name=\"number_of_elements_n\" id=\"field-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"5\" min=\"1\" 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class=\"reaction-count\" data-reaction-count=\"like\">0<\/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\">0<\/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\">1<\/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\">1<\/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>Stirling\u2019s Approximation 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\\[n! \\approx \\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^{n}\\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(n\\) \u2014 the number whose factorial is approximated (\\(n \\ge 1\\))<\/li>\n<li>\\(\\pi\\) \u2014 the mathematical constant pi (\u2248 3.14159)<\/li>\n<li>\\(e\\) \u2014 Euler's number (\u2248 2.71828)<\/li>\n<\/ul>\n<p>The formula replaces the product \\(1 \\cdot 2 \\cdot 3 \\cdots n\\) with a closed-form expression involving a square root and an exponential term, which can be evaluated in constant time no matter how large \\(n\\) is.<\/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>Stirling\u2019s Approximation \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> approximate the factorial of 5.<\/p>\n<p>\\[n! \\approx \\sqrt{2\\pi \\cdot 5}\\left(\\frac{5}{e}\\right)^{5}\\]<\/p>\n<p>\\[\\sqrt{10\\pi} \\approx 5.60499, \\qquad \\left(\\frac{5}{e}\\right)^{5} = \\frac{3125}{e^{5}} \\approx 21.05609\\]<\/p>\n<p>\\[5! \\approx 5.60499 \\times 21.05609 \\approx 118.019\\]<\/p>\n<p><strong>Result:<\/strong> \\(5! \\approx 1.180190 \\times 10^{2}\\). The exact value is \\(5! = 120\\), so the approximation underestimates it by about 1.65%.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Enter a positive integer in the <strong>Number of elements (n)<\/strong> field. The default value is 5.<\/li>\n<li>The result field labeled <strong>n! \u2248<\/strong> immediately shows the approximated factorial in scientific notation (for example, <code>1.180190e+2<\/code>).<\/li>\n<li>To estimate a different factorial, simply type a new value for \\(n\\); the result updates automatically.<\/li>\n<\/ol>\n<h2>What Is Stirling\u2019s Approximation?<\/h2>\n<p>A factorial \\(n!\\) is the product of all integers from 1 to \\(n\\). Factorials grow extremely fast: \\(10!\\) is already 3,628,800, and \\(100!\\) is a 158-digit number. Computing such values exactly requires arbitrary-precision arithmetic, and in many applications the exact digits are simply not needed. Stirling's approximation, named after the 18th-century Scottish mathematician James Stirling, gives a fast estimate of \\(n!\\) using only elementary operations.<\/p>\n<p>The approximation is remarkably accurate. The relative error is roughly \\(\\frac{1}{12n}\\), so it is about 0.8% for \\(n = 10\\), 0.1% for \\(n = 80\\), and vanishingly small for very large \\(n\\). This is why the formula is a cornerstone of asymptotic analysis: it converts products that grow factorially into expressions involving \\(n^{n}\\), \\(e^{n}\\), and \\(\\sqrt{n}\\), which are far easier to manipulate algebraically.<\/p>\n<p>Stirling's approximation appears throughout mathematics and science. It underlies the derivation of the normal approximation to the binomial distribution, the entropy formula in statistical mechanics and information theory, and the asymptotic behavior of the Gamma function \\(\\Gamma(n+1) = n!\\). It is also the standard tool for estimating binomial coefficients, which count the arrangements in combinatorial structures such as those generated by the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/leibniz-harmonic-triangle-generator\/\">Leibniz harmonic triangle<\/a> or the sums produced by <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/floyds-triangle-generator\/\">Floyd's triangle<\/a>.<\/p>\n<h2>Accuracy of the Approximation<\/h2>\n<table>\n<thead>\n<tr>\n<th>\\(n\\)<\/th>\n<th>Exact \\(n!\\)<\/th>\n<th>Stirling\u2019s approximation<\/th>\n<th>Relative error<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1<\/td>\n<td>1<\/td>\n<td>0.9221<\/td>\n<td>\u22127.79%<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>2<\/td>\n<td>1.9190<\/td>\n<td>\u22124.05%<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>120<\/td>\n<td>118.019<\/td>\n<td>\u22121.65%<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>3,628,800<\/td>\n<td>3,598,696<\/td>\n<td>\u22120.83%<\/td>\n<\/tr>\n<tr>\n<td>20<\/td>\n<td>2.4329\u00d710<sup>18<\/sup><\/td>\n<td>2.4228\u00d710<sup>18<\/sup><\/td>\n<td>\u22120.42%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that the approximation always underestimates the true value, and the error shrinks steadily as \\(n\\) grows. For most practical purposes the estimate is already excellent by \\(n = 20\\).<\/p>\n<h2>Limitations and Practical Tips<\/h2>\n<ul>\n<li><strong>Small n:<\/strong> the formula is intended for large \\(n\\). For \\(n &lt; 10\\) the error may be noticeable, so use the exact factorial when every digit matters.<\/li>\n<li><strong>Always a slight underestimate:<\/strong> the simple version used here omits correction factors; a more precise variant multiplies the result by \\(e^{1\/(12n)}\\).<\/li>\n<li><strong>Scientific notation output:<\/strong> results are displayed in exponential form (e.g., <code>9.332621e+157<\/code> for \\(n = 100\\)), which is the standard way to handle numbers too large for ordinary decimal notation.<\/li>\n<li><strong>Integer input only:<\/strong> enter a positive integer. For non-integer arguments, the generalization of the factorial is the Gamma function, which satisfies the same approximation for large arguments.<\/li>\n<li><strong>Order-of-magnitude work:<\/strong> when comparing growth rates, as with sequences built from an <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/arithmetic-progression\/\">arithmetic progression<\/a> versus factorial growth, Stirling's formula quickly shows that \\(n!\\) eventually dwarfs any polynomial or exponential \\(c^{n}\\).<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Estimate the factorial of a large number instantly with Stirling's approximation: enter n and get n! \u2248 \u221a(2\u03c0n)(n\/e)\u207f in scientific notation. Ideal for statistics, probability, and combinatorics.<\/p>\n","protected":false},"author":70,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[72],"class_list":["post-61475","post","type-post","status-publish","format-standard","hentry","category-mathematics","tag-arithmetic-calculators"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61475","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\/70"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=61475"}],"version-history":[{"count":2,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61475\/revisions"}],"predecessor-version":[{"id":66288,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61475\/revisions\/66288"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=61475"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=61475"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=61475"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}