{"id":156268,"date":"2022-03-26T01:34:07","date_gmt":"2022-03-25T20:04:07","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/binomial-theorem-class-11-definition-formula-properties-and-faqs\/"},"modified":"2025-06-23T16:53:22","modified_gmt":"2025-06-23T11:23:22","slug":"binomial-theorem-class-11-definition-formula-properties-and-faqs","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/binomial-theorem-class-11\/","title":{"rendered":"Binomial Theorem Class 11 \u2013 Definition, Formula, Properties and FAQs"},"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-class-11\/#About_Binomial_Theorem_Class_11_Chapter_8\" title=\"About Binomial Theorem Class 11 Chapter 8\">About Binomial Theorem Class 11 Chapter 8<\/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-class-11\/#NCERT_Solutions_for_Binomial_Theorem_Class_11_Maths_Chapter_8\" title=\"NCERT Solutions for Binomial Theorem Class 11 Maths Chapter 8\">NCERT Solutions for Binomial Theorem Class 11 Maths Chapter 8<\/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-class-11\/#Definition_of_Binomial_Theorem\" title=\"Definition of Binomial Theorem\">Definition of 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-class-11\/#Binomial_Expansion\" title=\"Binomial Expansion\">Binomial Expansion<\/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-class-11\/#Terms_in_the_Binomial_Expansion\" title=\"Terms in the Binomial Expansion\">Terms in the Binomial Expansion<\/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-class-11\/#What_is_a_Binomial_Expression\" title=\"What is a Binomial Expression?\">What is a Binomial Expression?<\/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-class-11\/#Topics_and_Subtopics_of_Binomial_Theorem\" title=\"Topics and Subtopics of Binomial Theorem\">Topics and Subtopics of Binomial Theorem<\/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-class-11\/#Binomial_Theorem_Formula\" title=\"Binomial Theorem Formula\">Binomial Theorem Formula<\/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-class-11\/#Properties_of_the_Binomial_Expansion_x_yn\" title=\"Properties of the Binomial Expansion (x + y)n\">Properties of the Binomial Expansion (x + y)n<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/binomial-theorem-class-11\/#FAQs\" title=\"FAQs\">FAQs<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"About_Binomial_Theorem_Class_11_Chapter_8\"><\/span>About Binomial Theorem Class 11 Chapter 8<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Binomial Theorem Class 11: The <a href=\"https:\/\/infinitylearn.com\/surge\/maths\/binomial-theorem-class-11\/\">binomial theorem<\/a> is a theorem in mathematics that states that the sum of two terms is the product of the first term and the binomial coefficient of the second term.<\/p>\n<p>Therefore the binomial theorem is usually written as:<\/p>\n<p>(x + y)n = xn + yn + (xn-1y + xn-2y2 + &#8230; + yn-1)<\/p>\n<p>Where:<\/p>\n<p>x is the first term<\/p>\n<p>n is the number of terms<\/p>\n<p>y is the second term<\/p>\n<p>The binomial theorem can be used to calculate the expansion of a binomial expression.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_for_Binomial_Theorem_Class_11_Maths_Chapter_8\"><\/span>NCERT Solutions for Binomial Theorem Class 11 Maths Chapter 8<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The Binomial Theorem is a mathematical theorem that describes the way in which a binomial expansion behaves. Therefore theorem is named for the mathematician and philosopher Gottfried Wilhelm Leibniz, who published it in 1676.<\/p>\n<p>The theorem states that, for any real number x and any positive integer n, the following equation holds true:<\/p>\n<p>(x + y)n = xn + yn + nxyn<\/p>\n<p>The theorem can expanded to cover complex numbers by taking the imaginary unit i to be equal to \u22121. In this case, the equation becomes:<\/p>\n<p>(x + y)n = xn + yn \u2212 nxyn<\/p>\n<p>The binomial theorem can proven using mathematical induction.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Definition_of_Binomial_Theorem\"><\/span>Definition of Binomial Theorem<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The binomial theorem is a mathematical theorem that states that the expansion of a binomial (that is, the sum of two terms) is a sum of terms in which each term is the product of a power of the binomial&#8217;s two factors. The theorem named for the mathematician and theologian Pierre de Fermat, who first stated it in 1654.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Binomial_Expansion\"><\/span>Binomial Expansion<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>In mathematics, binomial expansion is the expansion of a binomial (a sum of two terms) into a series of terms, each of which is a product of the two original terms. Therefore the expansion usually written using the symbols of mathematics, with the terms in the series arranged in descending powers of the variable x.<\/p>\n<ul>\n<li>For example, the expansion of (x + y)5 is<\/li>\n<li>The coefficients a k are called the binomial coefficients.<\/li>\n<li>The binomial expansion is an important technique in mathematics, and also has many applications. One of the most famous applications is the proof of the binomial theorem.<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"Terms_in_the_Binomial_Expansion\"><\/span>Terms in the Binomial Expansion<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The binomial theorem states that for any real number x and also any positive integer n,<\/p>\n<p>(x + y)n = xn + yn + (n \u2212 1)xn\u22121y + (n \u2212 2)xn\u22122y2 + (n \u2212 3)xn\u22123y3 + \u00b7\u00b7\u00b7<\/p>\n<p>The coefficients of xn, yn, xn\u22121y, xn\u22122y2, and xn\u22123y3 in this expansion are<\/p>\n<p>1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"What_is_a_Binomial_Expression\"><\/span>What is a Binomial Expression?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A binomial expression is an algebraic expression that contains two terms, separated by a plus or minus sign.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Topics_and_Subtopics_of_Binomial_Theorem\"><\/span>Topics and Subtopics of Binomial Theorem<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>Binomial theorem<\/strong><\/p>\n<p>-The binomial theorem states that for any real number &#8220;n&#8221; and also any positive integer &#8220;k&#8221;,<\/p>\n<p>(x + y)^k = x^k + y^k + kx^k y^k<\/p>\n<p>-This equation can used to calculate the value of a binomial expansion, which is an expression that contains powers of binomial.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Binomial_Theorem_Formula\"><\/span>Binomial Theorem Formula<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The binomial theorem is a mathematical formula that states that a binomial expansion of a positive integer power of a binomial is the sum of all the positive integer powers of the individual binomial coefficients.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Properties_of_the_Binomial_Expansion_x_yn\"><\/span>Properties of the Binomial Expansion (x + y)n<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ol>\n<li>The binomial expansion is symmetrical about the middle term.<\/li>\n<li>binomial expansion is linear.<\/li>\n<li>The binomial expansion is a polynomial.<\/li>\n<li>binomial expansion is unique.<\/li>\n<\/ol>\n<h2><span class=\"ez-toc-section\" id=\"FAQs\"><\/span>FAQs<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Q: What is the Binomial Theorem? A: The Binomial Theorem is a mathematical formula that allows us to expand expressions of the form (a+b)^n, where a and b are any two numbers and n is a positive integer.<\/p>\n<p>Q: What is the formula for the Binomial Theorem? A: (a+b)^n = C(n,0)<em>a^n<\/em>b^0 + C(n,1)*a^(n-1)*b^1 + C(n,2)*a^(n-2)*b^2 + &#8230; + C(n,n)<em>a^0<\/em>b^n, where C(n,k) = n!\/(k!(n-k)!) is the binomial coefficient.<\/p>\n<p>Q: What is a binomial coefficient? A: A binomial coefficient is a number that represents the number of ways to choose k items from a set of n items. The binomial coefficient is denoted by C(n,k) or (n choose k).<\/p>\n<p>Q: How do I calculate binomial coefficients? A: The binomial coefficient can be calculated using the formula C(n,k) = n!\/(k!(n-k)!), where n! is the factorial of n.<\/p>\n<p>Q: What is a factorial? A: The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers from 1 to n. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>About Binomial Theorem Class 11 Chapter 8 Binomial Theorem Class 11: The binomial theorem is a theorem in mathematics that [&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 Class 11","_yoast_wpseo_title":"Class 11 Binomial Theorem - Definition, Formula, Properties and FAQs","_yoast_wpseo_metadesc":"NCERT Solutions for Class 11 Maths Chapter 8 - Binomial Theorem. 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