Let (1 + x)(1 + x + x^2)(1 + x + x^2 + x^3),, ......,, (1 + x + x^2 + ..... + x^{30}) = a_0 + a₁x

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7.Binomial Theorem

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Let $(1 + x)(1 + x + x^2)(1 + x + x^2 + x^3)\,\, ......\,\,$$(1 + x + x^2 + ..... + x^{30}) = $$a_0 + a_1x + a_2x^2$ .....$+$ $a_{465}x^{465}$, then sum of $a_0 + a_2 + a_4 + ......... +$ is

A

$(31)!$

B

$\frac{(31)!}{2}$

C

$(30)!$

D

$\frac{(60)!}{2}$

Solution

put $x=1$

$(31) !=a_{0}+a_{1}+a_{2}+\ldots \ldots$ $(1)$

put $x=-1$

$0=a_{0}-a_{1}+a_{2}-\ldots \ldots \ldots$ $(2)$

$(1)+(2) \Rightarrow \frac{(31) !}{2}=a_{0}+a_{2}+a_{4}+\ldots \ldots .$

Standard 11

Mathematics

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