{"id":6471,"date":"2021-10-16T17:33:17","date_gmt":"2021-10-16T12:03:17","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6471"},"modified":"2021-11-20T01:33:12","modified_gmt":"2021-11-19T20:03:12","slug":"formula-for-binomial-theorem","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/formula-for-binomial-theorem\/","title":{"rendered":"Formula for Binomial Theorem"},"content":{"rendered":"

Here you will learn formula for binomial theorem of class 11 with examples.<\/p>\n

Let’s begin –<\/p>\n

Formula for Binomial Theorem<\/h2>\n

If x and a are real numbers, then for all n \\(\\in\\) N.<\/p>\n

\n

\\((x + a)^n\\) = \\(^{n}C_0 x^n a^0\\) + \\(^{n}C_1 x^{n – 1} a^1\\)\u00a0+ …………… + \\(^{n}C_r x^{n – r} a^r\\) + …………… + \\(^{n}C_{n – 1} x^1 a^{n – 1}\\) + \\(^{n}C_n x^0 a^n\\)<\/p>\n

i.e. \\((x + a)^n\\) = \\(\\sum_{r=0}^{n}\\) \\(^{n}C_{r} x^{n-r} a^{r}\\)<\/p>\n<\/blockquote>\n

Some Importants Result From Binomial Theorem<\/strong><\/h4>\n

(1)\u00a0<\/strong>We have.<\/p>\n

\n

\\((x + a)^n\\) = \\(\\sum_{r=0}^{n}\\) \\(^{n}C_{r} x^{n-r} a^{r}\\).<\/p>\n

or, \\((x + a)^n\\) = \\(^{n}C_0 x^n a^0\\) + \\(^{n}C_1 x^{n – 1} a^1\\) + …………… + \\(^{n}C_r x^{n – r} a^r\\) + …………… + \\(^{n}C_n x^0 a^n\\)<\/p>\n<\/blockquote>\n

Since r can have values from 0 to n, the total number of terms in the expansion is (n + 1).<\/p>\n

(2)<\/strong> The sum of the indices of x and a in each term is n.<\/p>\n

(3)<\/strong> Since \\(^{n}C_r\\) = \\(^{n}C_{n – r}\\)<\/p>\n

\n

\\(\\implies\\) \\(^{n}C_0\\) = \\(^{n}C_n\\), \\(^{n}C_1\\) = \\(^{n}C_{n-1}\\), \\(^{n}C_2\\) = \\(^{n}C_{n – 2}\\) = …….<\/p>\n<\/blockquote>\n

So, the coefficients of terms equidistant from the beginning and end are equal. These coefficients are known as binomial coefficients.<\/p>\n

(4)<\/strong> Replacing a by -a, we get<\/p>\n

\n

\\((x – a)^n\\) = \\(^{n}C_0 x^n a^0\\) – \\(^{n}C_1 x^{n – 1} a^1\\) + …………… + \\((-1)^r\\)\\(^{n}C_r x^{n – r} a^r\\) + …………… + \\((-1)^n\\)\\(^{n}C_n x^0 a^n\\)<\/p>\n

i.e.\u00a0 \\((x – a)^n\\) = \\(\\sum_{r=0}^{n}\\) \\((-1)^r\\) \\(^{n}C_{r} x^{n-r} a^{r}\\)<\/p>\n<\/blockquote>\n

Thus, the terms in the expansion of \\((x – a)^n\\) are alternatively positive and negative, the last term is positive or negative according as n is even or odd.<\/p>\n

(5)<\/strong> Putting x = 1 and a = x in the expansion of \\((x + a)^n\\), we get<\/p>\n

\n

\\((1 + x)^n\\) = \\(^{n}C_0\\) + \\(^{n}C_1 x\\) + \\(^{n}C_2 x^2\\) + ……… + \\(^{n}C_r x^r\\) + \\(^{n}C_n x^n\\)<\/p>\n

i.e.\u00a0 \\((1 + x)^n\\) = \\(\\sum_{r=0}^{n}\\) \\(^{n}C_r x^r\\)<\/p>\n<\/blockquote>\n

This is the expansion of \\((1 + x)^n\\) in ascending powers of x.<\/p>\n

(6)<\/strong> Putting a = 1 in the expansion of \\((x + a)^n\\), we get.<\/p>\n

\n

\\((1 + x)^n\\) = \\(^{n}C_0 x^n\\) + \\(^{n}C_1 x^{n-1}\\) + \\(^{n}C_2 x^{n-2}\\) + ……… + \\(^{n}C_r x^{n-r}\\) + \\(^{n}C_{n-1} x\\) + \\(^{n}C_n\\).<\/p>\n

i.e.\u00a0 \\((1 + x)^n\\) = \\(\\sum_{r=0}^{n}\\) \\(^{n}C_r x^{n-r}\\)<\/p>\n<\/blockquote>\n

This is the expansion of \\((1 + x)^n\\) in descending powers of x.<\/p>\n

(7)<\/strong> Putting x = 1 and a = -x in the expansion of \\((x + a)^n\\), we get<\/p>\n

\n

\\((1 – x)^n\\) = \\(^{n}C_0\\) – \\(^{n}C_1 x\\) + \\(^{n}C_2 x^2\\) + ……… + \\((-1)^r\\)\\(^{n}C_r x^r\\) + \\((-1)^n\\) \\(^{n}C_n x^n\\).<\/p>\n

i.e.\u00a0 \\((1 – x)^n\\) = \\(\\sum_{r=0}^{n}\\) \\((-1)^n\\) \\(^{n}C_r x^r\\)<\/p>\n<\/blockquote>\n

(8)<\/strong> The coefficient of (r + 1)th term in the expansion of \\((1 + x)^n\\) is \\(^{n}C_r\\).<\/p>\n

(9)<\/strong> The coefficient of \\(x^r\\) in the expansion of \\((1 + x)^n\\) is \\(^{n}C_r\\).<\/p>\n

Example<\/span><\/strong> : Expand \\((x^2 + 2a)^5\\) by binomial theorem.<\/p>\n

Solution<\/span><\/strong> : Using binomial theorem, we have<\/p>\n

\\((x^2 + 2a)^5\\) = \\(^5C_0 (x^2)^5 (2a)^0\\) + \\(^5C_1 (x^2)^4 (2a)^1\\) + \\(^5C_2 (x^2)^3 (2a)^2\\) + \\(^5C_3 (x^2)^2 (2a)^3\\) + \\(^5C_4 (x^2) (2a)^4\\) + \\(^5C_5 (x^2)^0 (2a)^5\\)<\/p>\n

= \\(x^{10}\\) + 5\\((x^8)(2a)\\) + 10\\((x^6)(4a^2)\\) + 10\\((x^4)(8a^3)\\) + 5\\((x^2)(16a^4)\\) + 325\\((a^5)\\)<\/p>\n

=\u00a0 \\(x^{10}\\) + 10\\(x^{8}a\\) + 40\\(x^{6}a^2\\) + 80\\(x^{4}a^3\\) + 80\\(x^{2}a^4\\) + 32\\(a^{5}\\)<\/p>\n

\n

Related Questions<\/h3>\n

By using binomial theorem, expand \\((1 + x + x^2)^3\\).<\/a><\/p>\n

Which is larger \\((1.01)^{1000000}\\) or 10,000?<\/a><\/p>\n\n\n

\n
Next – General Term in Binomial Expansion<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn formula for binomial theorem of class 11 with examples. Let’s begin – Formula for Binomial Theorem If x and a are real numbers, then for all n \\(\\in\\) N. \\((x + a)^n\\) = \\(^{n}C_0 x^n a^0\\) + \\(^{n}C_1 x^{n – 1} a^1\\)\u00a0+ …………… + \\(^{n}C_r x^{n – r} a^r\\) + …………… …<\/p>\n

Formula for Binomial Theorem<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[40],"tags":[173,172,174],"yoast_head":"\nFormula for Binomial Theorem - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn formula for binomial theorem of class 11 with examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/formula-for-binomial-theorem\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Formula for Binomial Theorem - 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