{"id":307,"date":"2019-08-15T17:50:52","date_gmt":"2019-08-15T17:50:52","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=307"},"modified":"2020-01-15T20:55:12","modified_gmt":"2020-01-15T20:55:12","slug":"binomial-theorem","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/precalculus\/binomial-theorem\/","title":{"rendered":"Binomial Theorem"},"content":{"rendered":"\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n\n\n<title>The Binomial Theorem &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>\n<!------------------------>\n\nWe know that \n\\begin{eqnarray*}\n\t(x+y)^0&amp;=&amp;1\\\\\n\t(x+y)^1&amp;=&amp;x+y\\\\\n\t(x+y)^2&amp;=&amp;x^2+2xy+y^2\n\\end{eqnarray*}\nand we can easily expand\n\\[(x+y)^3=x^3+3x^2y+3xy^2+y^3.\\]\nFor higher powers, the expansion gets very tedious by hand!\nFortunately, the Binomial Theorem gives us the expansion for any\npositive integer power of $(x+y)$:\n\n<\/p>\n\n\n\n<p>\n\nFor any positive integer $n$,\n\\[(x+y)^n=\\sum^n_{k=0} \\left(\\begin{array}{c}\n\tn\\\\\n\tk\n\\end{array}\\right)x^{n-k}y^k\\]\nwhere \n$$\n\\displaystyle\\left(\\begin{array}{c}\n\tn\\\\\n\tk\n\\end{array}\\right)=\\frac{(n)(n-1)(n-2)\\cdots(n-(k-1))}{k!}=\\frac{n!}{k!(n-k)!}.\n$$\n<\/p>\n\n\n<style>.kt-accordion-id_41d87a-64 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_41d87a-64 .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 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.kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_41d87a-64:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:before{background:#555555;}.kt-accordion-id_41d87a-64:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger{background:#555555;}.kt-accordion-id_41d87a-64:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_41d87a-64:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:before{background:#f2f2f2;}.kt-accordion-id_41d87a-64 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id_41d87a-64 .kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_41d87a-64:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_41d87a-64:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_41d87a-64:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_41d87a-64:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_41d87a-64:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_41d87a-64:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_41d87a-64:not( .kt-accodion-icon-style-basic ):not( 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.kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_41d87a-64 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_41d87a-64 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_41d87a-64:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_41d87a-64:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_41d87a-64:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_41d87a-64:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_41d87a-64:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_41d87a-64 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_41d87a-64 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_41d87a-64 kt-accordion-has-4-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_8bcffa-5e\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Proof by Induction<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n<title>Binomial Theorem, Proof by Induction<\/title>\n\n<center><font size=\"+2\">\nBinomial Theorem\n<\/font><\/center>\n\n\n\n<p>\n\nFor any positive integer $n$, \\[(x+y)^n=\\sum^n_{k=0} \\left(\\begin{array}{c} n\\\\ k \\end{array}\\right)x^{n-k}y^k.\\] \n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Proof by Induction:<\/h4>\n\n\n\n<p>\n\nFor $n=1$, \n\\[(x+y)^1=x+y=\\left(\\begin{array}{c}\n\t1\\\\\n\t0\n\\end{array}\\right)x^{1-0}y^0+\\left(\\begin{array}{c}\n\t1\\\\\n\t1\n\\end{array}\\right)x^{1-1}y^1=\\sum_{k=0}^1 \\left(\\begin{array}{c}\n\t1\\\\\n\tk\n\\end{array}\\right)x^{1-k}y^k.\\]\n\n<\/p>\n\n\n\n<p>\nSuppose $\\displaystyle (x+y)^{n-1}=\\sum^{n-1}_{k=0} \\left(\\begin{array}{c}\n\tn-1\\\\\n\tk\n\\end{array}\\right)x^{(n-1)-k}y^k.$\n\n<\/p>\n\n\n\n<p>\nConsider $(x+y)^n$.\n\\begin{eqnarray*}\n\t(x+y)^n&amp;=&amp;(x+y)(x+y)^{n-1}\\\\\n\t&amp;=&amp;(x+y)\\left[\\sum^{n-1}_{k=0} \\left(\\begin{array}{c}\n\t\tn-1\\\\\n\t\tk\n\t\\end{array}\\right)x^{(n-1)-k}y^k\\right]\\\\\n\t&amp;=&amp; \n\t\\sum^{n-1}_{k=0} \\left(\\begin{array}{c}\n\t\tn-1\\\\\n\t\tk\n\t\\end{array}\\right)x^{n-k}y^k + \\sum^{n-1}_{j=0} \\left(\\begin{array}{c}\n\t\tn-1\\\\\n\t\tj\n\t\\end{array}\\right)x^{(n-1)-j}y^{j+1}\\\\\n\t&amp;=&amp;\\sum^{n-1}_{k=0} \\left(\\begin{array}{c}\n\t\tn-1\\\\\n\t\tk\n\t\\end{array}\\right)x^{n-k}y^k+\\sum^{n-1}_{j=0} \\left(\\begin{array}{c}\n\t\tn-1\\\\\n\t\t(j+1)-1\n\t\\end{array}\\right)x^{n-(j+1)}y^{j+1}\\\\\n\t&amp;=&amp;\\sum^{n-1}_{k=0} \\left(\\begin{array}{c}\n\t\tn-1\\\\\n\t\tk\n\t\\end{array}\\right)x^{n-k}y^k+\\sum^{n}_{k=1} \\left(\\begin{array}{c}\n\t\tn-1\\\\\n\t\tk-1\n\t\\end{array}\\right)x^{n-k}y^{k}\\\\\n\t&amp;=&amp;\\sum^{n}_{k=0} \\left[\\left(\\begin{array}{c}\n\t\tn-1\\\\\n\t\tk\n\t\\end{array}\\right)x^{n-k}y^k\\right]-\\left(\\begin{array}{c}\n\t\tn-1\\\\\n\t\tn\n\t\\end{array}\\right)x^0y^n\\\\\n\t&amp;~&amp;+\\sum^{n}_{k=0} \\left[\\left(\\begin{array}{c}\n\t\tn-1\\\\\n\t\tk-1\n\t\\end{array}\\right)x^{n-k}y^{k}\\right]-\\left(\\begin{array}{c}\n\t\tn-1\\\\\n\t\t-1\n\t\\end{array}\\right)x^ny^0\\\\\n\t&amp;=&amp;\\sum^{n}_{k=0} \\left[\\left(\\begin{array}{c}\n\t\tn-1\\\\\n\t\tk\n\t\\end{array}\\right)+\\left(\\begin{array}{c}\n\t\tn-1\\\\\n\t\tk-1\n\t\\end{array}\\right)\\right]x^{n-k}y^k\\\\\n\t&amp;=&amp; \\sum^{n}_{k=0} \\left(\\begin{array}{c}\n\t\tn\\\\\n\t\tk\n\t\\end{array}\\right)x^{n-k}y^k\n\\end{eqnarray*}\nand the theorem is proved!\n\n<!------------------------>\n\n\n<\/p>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-3 kt-pane_750191-84\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Combinatorial Proof<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n<title>Binomial Theorem, Combinatorial Proof<\/title>\n\n<center><font size=\"+2\">\nBinomial Theorem\n<\/font><\/center>\n\n\n\n<p>\n<!------------------------>\n\nFor any positive integer $n$,\n\\[(x+y)^n=\\sum^n_{k=0} \\left(\\begin{array}{c}\n\tn\\\\\n\tk\n\\end{array}\\right)x^{n-k}y^k.\\]\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Combinatorial Proof:<\/h4>\n\n\n\n<p>\n\nSince\n\\[(x+y)^n=\\underbrace{(x+y)(x+y)\\cdots(x+y)}_{\\textrm{$n$ of these}}\\]\neach term in $(x+y)^n$ has the form $x^{n-k}y^k$ for some $k$ between\n$0$ and $n$, inclusive. \n\n<\/p>\n\n\n\n<p> The coefficient of $x^{n-k}y^k$ for a particular $k$ is just the number of ways to choose $k$ factors of $y$ from the $n$ factors of $(x+y)$, with factors of $x$ coming from the remaining $(n-k)$ factors.  The number of ways to choose $k$ objects from a collection of $n$ objects, without replacement and order not important, is just  $\\displaystyle \\left(\\begin{array}{c} n\\\\ k \\end{array}\\right)$. <\/p>\n\n\n\n<p>\nThus, the term $x^{n-k}y^k$ has coefficient \n$\\displaystyle \\left(\\begin{array}{c}\n\tn\\\\\n\tk\n\\end{array}\\right)$, so\n\\[(x+y)^n=\\sum^n_{k=0} \\left(\\begin{array}{c}\n\tn\\\\\n\tk\n\\end{array}\\right)x^{n-k}y^k.\\]\n\n<!------------------------>\n\n\n<\/p>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-4 kt-pane_cf2f5d-f7\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Connection to Pascal&#8217;s Triangle<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n<title>Connection to Pascal&#8217;s Triangle<\/title>\n<center><font size=\"+2\">\nConnection to Pascal&#8217;s Triangle\n<\/font><\/center>\n\n<center>\n$\n\\begin{array}{ccc}\n\t(x+y)^0&amp;=&amp;1\\\\\n\t(x+y)^1&amp;=&amp;1x+1y\\\\\n\t(x+y)^2&amp;=&amp;1x^2+2xy+1y^2\\\\\n\t(x+y)^3&amp;=&amp;1x^3+3x^2y+3xy^2+1y^3\\\\\n\t(x+y)^4&amp;=&amp;1x^4+4x^3y+6x^2y^2+4xy^3+1y^4\\\\\n\t(x+y)^5&amp;=&amp;1x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+1y^5\n\\end{array}\n$\n<\/center>\n\n\n\n<p>\n\n Do you see the connection??  \n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nBy the Binomial Theorem,\n\\begin{eqnarray*}\n\t(x+y)^3&amp;=&amp; \\sum_{k=0}^3 \\left(\\begin{array}{c}\n\t\t3\\\\\n\t\tk\n\t\\end{array}\\right)x^{3-k}y^k\\\\\n\t&amp;=&amp; \\left(\\begin{array}{c}\n\t\t3\\\\\n\t\t0\n\t\\end{array}\\right)x^3+\\left(\\begin{array}{c}\n\t\t3\\\\\n\t\t1\n\t\\end{array}\\right)x^2y+\\left(\\begin{array}{c}\n\t\t3\\\\\n\t\t2\n\t\\end{array}\\right)xy^2+\\left(\\begin{array}{c}\n\t\t3\\\\\n\t\t3\n\t\\end{array}\\right)y^3\\\\\n\t&amp;=&amp; x^3+3x^2y+3xy^2+y^3\n\\end{eqnarray*}\nas expected.\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Extensions of the Binomial Theorem<\/h4>\n\n\n\n<p>\n\nA useful special case of the Binomial Theorem is\n\\[(1+x)^n=\\sum^n_{k=0} \\left(\\begin{array}{c}\n\tn\\\\\n\tk\n\\end{array}\\right)x^k\\]\nfor any positive integer $n$, which is just the Taylor series for\n$(1+x)^n$.  \n\n<\/p>\n\n\n\n<p> This formula can be extended to all real powers $\\alpha$: \\[(1+x)^{\\alpha}=\\sum^{\\infty}_{k=0} \\left(\\begin{array}{c} \\alpha\\\\ k \\end{array}\\right)x^k\\] for any real number $\\alpha$, where  \\[\\left(\\begin{array}{c} \\alpha\\\\ k \\end{array}\\right)=\\frac{(\\alpha)(\\alpha -1)(\\alpha -2)\\cdots (\\alpha -(k-1))}{k!}=\\frac{\\alpha !}{k!(\\alpha -k)!}.\\] Notice that the formula now gives an infinite series: when $\\alpha=n$ is a positive integer, all but the first $(n+1)$ terms are $0$ since after this $n-n$ ($=0$) appears in each numerator.<\/p>\n\n\n\n<p>\nThis expansion is very useful for approximating $(1+x)^{\\alpha}$ for\n$|x|\\ll 1$:\n\\[(1+x)^\\alpha =1+\\alpha x+ \\frac{\\alpha(\\alpha\n-1)}{2!}x^2+\\frac{\\alpha (\\alpha -1)(\\alpha -2)}{3!}x^3+\\cdots.\\]\nBut for $|x|\\ll 1$, higher powers of $x$ get small very quickly, so\n$(1+x)^{\\alpha}$ can be approximated to any accuracy we need by\ntruncating the series after a finite number of terms.\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nFor $|x|\\ll 1$,\n\\begin{eqnarray*}\n\t(1+x)^{5\/2}&amp;\\approx&amp; 1+\\frac{5}{2}x,\\\\\n\t(1-2x)^{100}&amp;\\approx&amp; 1-200x,\\\\\n\t(1+x^2)^{-3}&amp;\\approx&amp; 1-3x^2.\n\\end{eqnarray*}\n\nThis type of reasoning is useful in investigating what happens when a\nphysical system is perturbed slightly, introducing a new very small\nterm $x$.\n\n<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">Key Concepts<\/h4>\n\n\n\n<h6 class=\"wp-block-heading\">\n<b>Binomial Theorem<\/b>\n<\/h6>\n\n\n\n<p>\n\nFor any positive integer $n$,\n$$\n(x+y)^n = \\sum_{k=0}^n \\left(\\begin{array}{c} n \\\\ k \\end{array}\\right) x^{n-k}\ny^k\n$$\nwhere\n$$\n\\left(\\begin{array}{c} n \\\\ k \\end{array}\\right) = \n\\frac{n(n-1)(n-2)\\cdots(n-k+1)}{k!} = \\frac{n!}{k!(n-k)!}.\n$$\n\n<!------------------------>\n\n\n<br>\n\n<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\n\n[<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ1910\/\">I&#8217;m ready to take the quiz.<\/a>]\n[<a href=\"#top\">I need to review more.<\/a>]<br>\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The Binomial Theorem &#8211; HMC Calculus Tutorial We know that \\begin{eqnarray*} (x+y)^0&amp;=&amp;1\\\\ (x+y)^1&amp;=&amp;x+y\\\\ (x+y)^2&amp;=&amp;x^2+2xy+y^2 \\end{eqnarray*} and we can easily expand \\[(x+y)^3=x^3+3x^2y+3xy^2+y^3.\\] For higher powers, the expansion gets very tedious by hand! Fortunately, the Binomial Theorem gives us the expansion for any positive integer power of $(x+y)$: For any positive integer $n$, \\[(x+y)^n=\\sum^n_{k=0} \\left(\\begin{array}{c} n\\\\ k&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":55,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-307","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/307","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/comments?post=307"}],"version-history":[{"count":8,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/307\/revisions"}],"predecessor-version":[{"id":1162,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/307\/revisions\/1162"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/55"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=307"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=307"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}