Binomial Theorem And Its Simple Applications Question 162
Question: The sum of the coefficients of even power of x in the expansion of $ {{(1+x+x^{2}+x^{3})}^{5}} $ is
[EAMCET 1988]
Options:
A) 256
B) 128
C) 512
D) 64
Show Answer
Answer:
Correct Answer: C
Solution:
- $ {{(1+x+x^{2}+x^{3})}^{5}}={{(1+x)}^{5}}{{(1+x^{2})}^{5}} $
$ =(1+5x+10x^{2}+10x^{3}+5x^{4}+x^{5}) $
$ \times (1+5x^{2}+10x^{4}+10x^{6}+5x^{8}+x^{10}) $
Therefore the required sum of coefficients $ =(1+10+5){{.2}^{5}}=16\times 32=512 $
Note: $ 2^{n}=2^{5} $ = Sum of all the binomial coefficients in the 2nd bracket in which all the powers of x are even.
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