{"id":162,"date":"2025-02-13T22:44:29","date_gmt":"2025-02-13T22:44:29","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/binomial-theorem\/"},"modified":"2026-03-26T19:18:48","modified_gmt":"2026-03-26T19:18:48","slug":"binomial-theorem","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/binomial-theorem\/","title":{"raw":"Binomial Theorem: Learn It 1","rendered":"Binomial Theorem: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Use the Binomial Theorem to expand a binomial.<\/li>\r\n \t<li>Use the Binomial Theorem to find a specified term of a binomial expansion.<\/li>\r\n<\/ul>\r\n<\/section>A polynomial with two terms is called a binomial. We already know how to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming.\r\n\r\nThe <strong>Binomial Theorem<\/strong> is a powerful tool in counting because it allows us to expand expressions raised to a power, which can then be used to solve problems involving combinations. Specifically, it helps in counting the number of ways to choose a subset of items from a larger set when the order doesn\u2019t matter.\r\n\r\nLet's discuss a shortcut way that will allow us to find [latex](x+y)^n[\/latex] <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">without multiplying the binomial by itself [latex]n[\/latex] times!<\/span>\r\n<h2>Binomial Coefficients<\/h2>\r\nHave you ever noticed that there's a pattern to the coefficients when you expand [latex](x+y)^n[\/latex]?\r\n\r\nThese coefficients, known as <strong>binomial coefficients<\/strong>, follow a specific pattern that appears in Pascal's Triangle. Each coefficient represents the number of ways to choose a certain number of terms from the binomial expansion.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>binomial coefficients<\/h3>\r\nIf [latex]n[\/latex] and [latex]r[\/latex] are integers greater than or equal to [latex]0[\/latex] with [latex]n\\ge r[\/latex], then the <strong>binomial coefficient<\/strong> is\r\n<p style=\"text-align: center;\">[latex]\\left(\\begin{gathered}n\\\\ r\\end{gathered}\\right)=C\\left(n,r\\right)=\\dfrac{n!}{r!\\left(n-r\\right)!}[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find each binomial coefficient.\r\n<ol>\r\n \t<li>[latex]\\left(\\begin{gathered}5\\\\ 3\\end{gathered}\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left(\\begin{gathered}9\\\\ 2\\end{gathered}\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left(\\begin{gathered}9\\\\ 7\\end{gathered}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"286266\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"286266\"]\r\n\r\nUse the formula to calculate each binomial coefficient. You can also use the [latex]{n}_{}{C}_{r}[\/latex] function on your calculator.\r\n<p style=\"text-align: center;\">[latex]\\left(\\begin{gathered}n\\\\ r\\end{gathered}\\right)=C\\left(n,r\\right)=\\dfrac{n!}{r!\\left(n-r\\right)!}[\/latex]<\/p>\r\n\r\n<ol>\r\n \t<li>[latex]\\left(\\begin{gathered}5\\\\ 3\\end{gathered}\\right)=\\dfrac{5!}{3!\\left(5 - 3\\right)!}=\\dfrac{5\\cdot 4\\cdot 3!}{3!2!}=10[\/latex]<\/li>\r\n \t<li>[latex]\\left(\\begin{gathered}9\\\\ 2\\end{gathered}\\right)=\\dfrac{9!}{2!\\left(9 - 2\\right)!}=\\dfrac{9\\cdot 8\\cdot 7!}{2!7!}=36[\/latex]<\/li>\r\n \t<li>[latex]\\left(\\begin{gathered}9\\\\ 7\\end{gathered}\\right)=\\dfrac{9!}{7!\\left(9 - 7\\right)!}=\\dfrac{9\\cdot 8\\cdot 7!}{7!2!}=36[\/latex]<\/li>\r\n<\/ol>\r\n<strong>Analysis of the Solution<\/strong>\r\n\r\nNotice that we obtained the same result for parts (b) and (c). If you look closely at the solution for these two parts, you will see that you end up with the same two factorials in the denominator, but the order is reversed, just as with combinations.\r\n<p style=\"text-align: center;\">[latex]\\left(\\begin{gathered}n\\\\ r\\end{gathered}\\right)=\\left(\\begin{gathered}n\\\\ n-r\\end{gathered}\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322002[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322003[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322004[\/ohm_question]<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">The patterns that emerge from calculating binomial coefficients and that are present in Pascal's Triangle are handy and should be memorized over time as mathematical facts much in the same way that you just \"know\" [latex]4[\/latex] and [latex]3[\/latex] make [latex]7[\/latex]. Of course, that will take a lot of time and patient practice. If you are continuing in mathematics beyond this course, it will be well worth the effort.<\/section><\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Use the Binomial Theorem to expand a binomial.<\/li>\n<li>Use the Binomial Theorem to find a specified term of a binomial expansion.<\/li>\n<\/ul>\n<\/section>\n<p>A polynomial with two terms is called a binomial. We already know how to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming.<\/p>\n<p>The <strong>Binomial Theorem<\/strong> is a powerful tool in counting because it allows us to expand expressions raised to a power, which can then be used to solve problems involving combinations. Specifically, it helps in counting the number of ways to choose a subset of items from a larger set when the order doesn\u2019t matter.<\/p>\n<p>Let&#8217;s discuss a shortcut way that will allow us to find [latex](x+y)^n[\/latex] <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">without multiplying the binomial by itself [latex]n[\/latex] times!<\/span><\/p>\n<h2>Binomial Coefficients<\/h2>\n<p>Have you ever noticed that there&#8217;s a pattern to the coefficients when you expand [latex](x+y)^n[\/latex]?<\/p>\n<p>These coefficients, known as <strong>binomial coefficients<\/strong>, follow a specific pattern that appears in Pascal&#8217;s Triangle. Each coefficient represents the number of ways to choose a certain number of terms from the binomial expansion.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>binomial coefficients<\/h3>\n<p>If [latex]n[\/latex] and [latex]r[\/latex] are integers greater than or equal to [latex]0[\/latex] with [latex]n\\ge r[\/latex], then the <strong>binomial coefficient<\/strong> is<\/p>\n<p style=\"text-align: center;\">[latex]\\left(\\begin{gathered}n\\\\ r\\end{gathered}\\right)=C\\left(n,r\\right)=\\dfrac{n!}{r!\\left(n-r\\right)!}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find each binomial coefficient.<\/p>\n<ol>\n<li>[latex]\\left(\\begin{gathered}5\\\\ 3\\end{gathered}\\right)[\/latex]<\/li>\n<li>[latex]\\left(\\begin{gathered}9\\\\ 2\\end{gathered}\\right)[\/latex]<\/li>\n<li>[latex]\\left(\\begin{gathered}9\\\\ 7\\end{gathered}\\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q286266\">Show Solution<\/button><\/p>\n<div id=\"q286266\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the formula to calculate each binomial coefficient. You can also use the [latex]{n}_{}{C}_{r}[\/latex] function on your calculator.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(\\begin{gathered}n\\\\ r\\end{gathered}\\right)=C\\left(n,r\\right)=\\dfrac{n!}{r!\\left(n-r\\right)!}[\/latex]<\/p>\n<ol>\n<li>[latex]\\left(\\begin{gathered}5\\\\ 3\\end{gathered}\\right)=\\dfrac{5!}{3!\\left(5 - 3\\right)!}=\\dfrac{5\\cdot 4\\cdot 3!}{3!2!}=10[\/latex]<\/li>\n<li>[latex]\\left(\\begin{gathered}9\\\\ 2\\end{gathered}\\right)=\\dfrac{9!}{2!\\left(9 - 2\\right)!}=\\dfrac{9\\cdot 8\\cdot 7!}{2!7!}=36[\/latex]<\/li>\n<li>[latex]\\left(\\begin{gathered}9\\\\ 7\\end{gathered}\\right)=\\dfrac{9!}{7!\\left(9 - 7\\right)!}=\\dfrac{9\\cdot 8\\cdot 7!}{7!2!}=36[\/latex]<\/li>\n<\/ol>\n<p><strong>Analysis of the Solution<\/strong><\/p>\n<p>Notice that we obtained the same result for parts (b) and (c). If you look closely at the solution for these two parts, you will see that you end up with the same two factorials in the denominator, but the order is reversed, just as with combinations.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(\\begin{gathered}n\\\\ r\\end{gathered}\\right)=\\left(\\begin{gathered}n\\\\ n-r\\end{gathered}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322002\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322002&theme=lumen&iframe_resize_id=ohm322002&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322003\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322003&theme=lumen&iframe_resize_id=ohm322003&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322004\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322004&theme=lumen&iframe_resize_id=ohm322004&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">The patterns that emerge from calculating binomial coefficients and that are present in Pascal&#8217;s Triangle are handy and should be memorized over time as mathematical facts much in the same way that you just &#8220;know&#8221; [latex]4[\/latex] and [latex]3[\/latex] make [latex]7[\/latex]. Of course, that will take a lot of time and patient practice. If you are continuing in mathematics beyond this course, it will be well worth the effort.<\/section>\n<\/div>\n","protected":false},"author":6,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":513,"module-header":"learn_it","content_attributions":[{"type":"original","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/162"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/162\/revisions"}],"predecessor-version":[{"id":6053,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/162\/revisions\/6053"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/513"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/162\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=162"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=162"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=162"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=162"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}