If {(1 + x)^n} = {C_0} + {C₁}x + {C₂}{x^2} + .... + {Cₙ}{x^n} , then {C

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7.Binomial Theorem

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If ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + .... + {C_n}{x^n}$, then ${C_0}{C_2} + {C_1}{C_3} + {C_2}{C_4} + {C_{n - 2}}{C_n}$ equals

$\frac{{(2n)!}}{{(n + 1)!(n + 2)!}}$

$\frac{{(2n)!}}{{(n - 2)!(n + 2)!}}$

$\frac{{(2n)!}}{{(n)!(n + 2)!}}$

$\frac{{(2n)!}}{{(n - 1)!(n + 2)!}}$

Solution

(b) We have, ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2}…. + {C_n}{x^n}$

${\left( {1 + \frac{1}{x}} \right)^n} = {C_0} + {C_1}.\frac{1}{x} + {C_2}.\frac{1}{{{x^2}}} + … + {C_n}\left( {\frac{1}{{{x^n}}}} \right)$

on multiplying both expansions, we get

$\frac{{{{(1 + x)}^{2n}}}}{{{x^n}}} = \sum {C_0^2 + x\sum {{C_0}{C_1} + {x^2}\sum {{C_0}{C_2} + ….} } } $$ + {x^r}\sum {{C_0}{C_r} + …..} $

The various sigma are the coefficient of ${x^0},x,{x^2},…..,{x^r}$ in $L.H.S.$ $\frac{{{{(1 + x)}^{2n}}}}{{{x^n}}}$ or coefficient of ${x^n},{x^{n + 1}},{x^{n + 2}},…..,{x^{n + r}}$ in the expansion of ${(1 + x)^{2n}}$ which occur in ${T_{n + 1,}}{T_{n + 2}},….$ and are

$^{2n}{C_n}{,^{2n}}{C_{n + 1}}{,^{2n}}{C_{n + 2}}{….^{2n}}{C_{n + r}}$etc.

$^{\,2n}{C_{n + 2}} = \frac{{(2\,n\,)\,!}}{{(n – 2)\,!\,(n + 2)\,!}}$

Standard 11

Mathematics

What is the coefficient of $x^{100}$ in $(1 + x + x^2 + x^3 +…. + x^{100})^3$ ?

normal

The sum to $(n + 1)$ terms of the series $\frac{{{C_0}}}{2} – \frac{{{C_1}}}{3} + \frac{{{C_2}}}{4} – \frac{{{C_3}}}{5} + …$ is

normal

Let $[ x ]$ denote greatest integer less than or equal to $x .$ If for $n \in N ,\left(1-x+x^{3}\right)^{n}=\sum_{j=0}^{3 n} a_{j} x^{j}$, then $\sum_{j=0}^{\left[\frac{3 n}{2}\right]} a_{2 j}+4 \sum_{j=0}^{\left[\frac{3 n-1}{2}\right]} a_{2 j+1}$ is equal to

normal

(JEE MAIN-2021)

If $(1 + x) (1 + x + x^2) (1 + x + x^2 + x^3) …… (1 + x + x^2 + x^3 + …… + x^n)$

$\equiv a_0 + a_1x + a_2x^2 + a_3x^3 + …… + a_mx^m$ then $\sum\limits_{r\, = \,0}^m {\,\,{a_r}}$ has the value equal to

normal

Let $n$ be an odd integer. If $\sin n\theta = \sum\limits_{r = 0}^n {{b_r}{{\sin }^r}\theta } $ for every value of $\theta $, then

medium

If ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + .... + {C_n}{x^n}$, then ${C_0}{C_2} + {C_1}{C_3} + {C_2}{C_4} + {C_{n - 2}}{C_n}$ equals

Solution

Similar Questions

What is the coefficient of $x^{100}$ in $(1 + x + x^2 + x^3 +…. + x^{100})^3$ ?

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Let $[ x ]$ denote greatest integer less than or equal to $x .$ If for $n \in N ,\left(1-x+x^{3}\right)^{n}=\sum_{j=0}^{3 n} a_{j} x^{j}$, then $\sum_{j=0}^{\left[\frac{3 n}{2}\right]} a_{2 j}+4 \sum_{j=0}^{\left[\frac{3 n-1}{2}\right]} a_{2 j+1}$ is equal to

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If $(1 + x) (1 + x + x^2) (1 + x + x^2 + x^3) …… (1 + x + x^2 + x^3 + …… + x^n)$

$\equiv a_0 + a_1x + a_2x^2 + a_3x^3 + …… + a_mx^m$ then $\sum\limits_{r\, = \,0}^m {\,\,{a_r}}$ has the value equal to