{"id":154226,"date":"2022-03-25T23:18:00","date_gmt":"2022-03-25T17:48:00","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/binomial-theorem-for-positive-integral-indices\/"},"modified":"2025-06-23T16:52:17","modified_gmt":"2025-06-23T11:22:17","slug":"binomial-theorem-for-positive-integral-indices","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/binomial-theorem-for-positive-integral-indices\/","title":{"rendered":"Binomial Theorem for Positive Integral Indices"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/binomial-theorem-for-positive-integral-indices\/#Introduction_to_Binomial_Theorem\" title=\"Introduction to Binomial Theorem\">Introduction to Binomial Theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/binomial-theorem-for-positive-integral-indices\/#The_Binomial_Theorem_is_different_from_a_Binomial_Distribution\" title=\"The Binomial Theorem is different from a Binomial Distribution\">The Binomial Theorem is different from a Binomial Distribution<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/binomial-theorem-for-positive-integral-indices\/#How_do_you_Apply_the_Binomial_Theorem\" title=\"How do you Apply the Binomial Theorem?\">How do you Apply the Binomial Theorem?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/binomial-theorem-for-positive-integral-indices\/#Application_in_Real-World_Situations_of_The_Binomial_Theorem\" title=\"Application in Real-World Situations of The Binomial Theorem\">Application in Real-World Situations of The Binomial Theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/binomial-theorem-for-positive-integral-indices\/#The_Binomial_Theorem_has_many_important_topics\" title=\"The Binomial Theorem has many important topics.\">The Binomial Theorem has many important topics.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/binomial-theorem-for-positive-integral-indices\/#Many_interesting_Properties_of_the_Binomial_Theorem\" title=\"Many interesting Properties of the Binomial Theorem\">Many interesting Properties of the Binomial Theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/binomial-theorem-for-positive-integral-indices\/#What_is_the_statement_of_Binomial_Theorem_for_Positive_Integral_Indices\" title=\"What is the statement of Binomial Theorem for Positive Integral Indices &#8211;\">What is the statement of Binomial Theorem for Positive Integral Indices &#8211;<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/binomial-theorem-for-positive-integral-indices\/#Proof_of_Binomial_Theorem\" title=\"Proof of Binomial Theorem &#8211;\">Proof of Binomial Theorem &#8211;<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/binomial-theorem-for-positive-integral-indices\/#Formula_for_Pascals_Triangle\" title=\"Formula for Pascal&#8217;s Triangle &#8211;\">Formula for Pascal&#8217;s Triangle &#8211;<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Introduction_to_Binomial_Theorem\"><\/span>Introduction to Binomial Theorem<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The binomial theorem is a mathematical theorem that states that for every positive integer n, the sum of the binomial coefficients of the first n positive integers is equal to 2n. The binomial theorem can be written using the summation symbol as:<\/p>\n<p>The binomial theorem is also known as the binomial expansion.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"The_Binomial_Theorem_is_different_from_a_Binomial_Distribution\"><\/span>The Binomial Theorem is different from a Binomial Distribution<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>in that the Binomial Theorem is an equation that can be used to calculate the probability of a specific outcome, while a Binomial Distribution is a chart that shows the probability of different outcomes.<\/p>\n<p>The Binomial Theorem is an equation that can be used to calculate the probability of a specific outcome. The equation is as follows:<\/p>\n<p>P(x) = (n choose x) px qn-x<\/p>\n<p>In this equation, &#8220;p&#8221; is the probability of success, &#8220;x&#8221; is the number of successes, &#8220;n&#8221; is the number of trials, and &#8220;q&#8221; is the probability of failure. This equation can be used to calculate the probability of any specific outcome.<\/p>\n<p>A Binomial Distribution is a chart that shows the probability of different outcomes. The chart is set up so that the x-axis represents the number of successes, and the y-axis represents the probability of each outcome. This chart can be used to help predict the probability of a specific outcome, but it cannot be used to calculate the probability of a specific outcome.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"How_do_you_Apply_the_Binomial_Theorem\"><\/span>How do you Apply the Binomial Theorem?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The binomial theorem states that if $x$ is a real number and $n$ is a positive integer, then<\/p>\n<p>$(x+y)^n=\\sum_{k=0}^{n}n\\binom{n}{k}x^{k}y^{n-k}$<\/p>\n<p>To apply the binomial theorem, you need to know how to calculate the sum of a series. In general, the sum of a series is given by<\/p>\n<p>$\\sum_{n=1}^{\\infty}nx^{n}$<\/p>\n<p>To calculate the sum of a series, you need to know the formula for the sum of a geometric series. The formula for the sum of a geometric series is<\/p>\n<p>$\\sum_{n=1}^{\\infty}rx^{n}=\\frac{1}{1-r}$<\/p>\n<p>Once you know the sum of a geometric series, you can use it to calculate the sum of a series using the binomial theorem. To do this, you need to use the following formula:<\/p>\n<p>$\\sum_{n=1}^{\\infty}nx^{n}=\\frac{1}{1-r}-\\frac{r}{1-r}+\\frac{r^2}{1-r^2}-\\frac{r^3}{1-r^3}+\\cdots$<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Application_in_Real-World_Situations_of_The_Binomial_Theorem\"><\/span>Application in Real-World Situations of The Binomial Theorem<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>In real-world situations, the binomial theorem can be applied in a number of different ways. For example, it can be used to calculate probabilities, to find the sum of a series, or to approximate certain functions. Additionally, the binomial theorem can be used to calculate certain properties of Pascal&#8217;s triangle.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"The_Binomial_Theorem_has_many_important_topics\"><\/span>The Binomial Theorem has many important topics.<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>These include the coefficient of xk, binomial expansions, and Pascal&#8217;s Triangle.<\/p>\n<p>The coefficient of xk is the number of ways that k can be chosen from a set of n elements. For example, the coefficient of x3 is 3 because there are 3 ways to choose 3 elements from a set of 10 elements.<\/p>\n<p>Binomial expansions are a way to calculate the terms of a polynomial. They are often used to find the solutions to certain types of equations.<\/p>\n<p>Pascal&#8217;s Triangle is a triangular array of numbers that can be used to calculate binomial expansions. It is named after Blaise Pascal, who studied it in the 17th century.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Many_interesting_Properties_of_the_Binomial_Theorem\"><\/span>Many interesting Properties of the Binomial Theorem<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The binomial theorem states that for any real number x and any positive integer n,<\/p>\n<p>(x + y)n = xn + yn + nxyn<\/p>\n<p>There are a few interesting properties of the binomial theorem that are worth mentioning.<\/p>\n<p>First, the binomial theorem is commutative. This means that<\/p>\n<p>(x + y)n = xn + yn<\/p>\n<p>For any real number x and any positive integer n, the order of the terms in the binomial expansion does not affect the result.<\/p>\n<p>Second, the binomial theorem is associative. This means that<\/p>\n<p>(x + y) + z = x + (y + z)<\/p>\n<p>For any real numbers x, y, and z, the order of the addends in a binomial expansion does not affect the result.<\/p>\n<p>Finally, the binomial theorem is distributive. This means that<\/p>\n<p>x(y + z) = xy + xz<\/p>\n<p>For any real numbers x, y, and z, the product of two binomials is the sum of the products of the binomials and the individual terms.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"What_is_the_statement_of_Binomial_Theorem_for_Positive_Integral_Indices\"><\/span>What is the statement of Binomial Theorem for Positive Integral Indices &#8211;<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The Binomial Theorem states that for every positive integral index n, there is a polynomial of degree n called the binomial coefficient polynomial, whose coefficient of xn is the binomial coefficient.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Proof_of_Binomial_Theorem\"><\/span>Proof of Binomial Theorem &#8211;<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>proof<\/p>\n<p>The Binomial Theorem is a theorem that states that for any real number x and any integer n, the following equation is true:<\/p>\n<p>(x + y)n = xn + yn + nxyn<\/p>\n<p>Proof:<\/p>\n<p>We will show that the left-hand side of the equation is equal to the right-hand side.<\/p>\n<p>To begin, we will show that the left-hand side is equal to xn + yn.<\/p>\n<p>We will use the following fact:<\/p>\n<p>(x + y)n = xn + yn + nxyn<\/p>\n<p>We can rewrite this equation as follows:<\/p>\n<p>xn + yn = (x + y)n &#8211; nxyn<\/p>\n<p>We can then subtract yn from both sides of the equation:<\/p>\n<p>xn + yn &#8211; yn = xn<\/p>\n<p>We can then divide both sides of the equation by n:<\/p>\n<p>xn \/ n = xn<\/p>\n<p>We have shown that xn + yn is equal to xn.<\/p>\n<p>Next, we will show that yn is equal to nxyn.<\/p>\n<p>We will use the following fact:<\/p>\n<p>(x + y)n = xn + yn + nxyn<\/p>\n<p>We can rewrite this equation as follows:<\/p>\n<p>xn + yn = (x + y)n &#8211; nxyn<\/p>\n<p>We can then subtract x<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Formula_for_Pascals_Triangle\"><\/span>Formula for Pascal&#8217;s Triangle &#8211;<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Binomial Theorem<\/p>\n<p>\\begin{align}<br \/>\n&amp;\\binom{n}{k} =\\frac{n!}{k!(n-k)!}\\\\<br \/>\n&amp;\\binom{n}{k} =\\frac{n(n-1)(n-2)\\dots (n-k+1)}{k(k-1)(k-2)\\dots (k+1)}\\\\<br \/>\n&amp;\\binom{n}{k} =\\frac{n(n-1)\\dots (n-k+1)}{k!}\\\\<br \/>\n&amp;\\binom{n}{k} =\\frac{n!}{k!(n-k)!}=\\frac{n!}{(k-1)!(n-k)!}\\\\<br \/>\n&amp;\\binom{n}{k} =\\frac{n!}{k!(n-k)!}=\\frac{n!}{k!}+\\frac{n!}{(k-1)!}\\\\<br \/>\n&amp;\\binom{n}{k} =\\frac{n!}{k!(n-k)!}=\\frac{n!}{k!}+\\frac{n!}{k-1}\\\\<br \/>\n&amp;\\binom{n}{k} =\\frac{n!}{k!(n-k)!}=\\frac{n!}{k!}+\\<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Introduction to Binomial Theorem The binomial theorem is a mathematical theorem that states that for every positive integer n, the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Binomial Theorem for Positive Integral Indices","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Learn about Binomial Theorem for Positive Integral Indices topic of Maths in details explained by subject experts on infinitylearn.com","custom_permalink":"maths\/binomial-theorem-for-positive-integral-indices\/"},"categories":[13],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - 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