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Question

The sum of the coefficients of even power of x in the expansion of (1+x+x2+x3)5 is

A
256
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B
128
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C
512
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D
64
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Solution

The correct option is C 512
(1+x+x2+x3)5=a0+a1x+a2x2+a3x3++a15x15
Putting x=1 and x=1 alternatively, we have
a0+a1+a2+a3++a15=45a0a1+a2a3+a15=0
Adding (1) and (2), we have 2(a0+a2+a4++a14)=45
a0+a2+a4++a14=29=512

Alternate Solution:
(1+x+x2+x3)5=(1+x)5(1+x2)5
Now (1+x2)5 have only even powers of x
so, the sum will be 25
and sum of even power coefficient in (1+x)5 will be 251=24
Hence required sum is
24×25=29=512

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