The coefficient of `x^9` in the expansion of `(1+x)(16 x^2)(1+x^3)(1+x^(100))` is

Options

A. 2
B. 6
C. 9
D. 8

Answer

d

Explanation

We observe that `x^(9)` can be formed in the following
8 ways :
`x^(9+0), x^(1+8), x^(2 +7), x ^(3 +6), x^(4 +5), x^(1 + 3 + 5), x^(1 + 2 + 6), x ^(2 + 3 + 4)` and the coefficients in each case is 1.
`therefore ` Coefficients of `x^(9) = underset(("8 - times"))(1 + 1 + ... + 1 = 8)` ALITER The coefficients of `x^(9)` is the number of ways in which the
sum of the power of x is 9 as listed below.
`x^(9) , x^(3) xxx^(8), x^(7), x^(3)xxx^(6), x^(4) xxx^(5), x^(1) xxx^(2)xxx^(6), x^(1) xxx^(3)xxx^(5),x^(2) xx^(3)xxx^(4)`.
Clearly, these are 8 ways.