The coefficient of x^18 in the product (1+x)(1-x)^10(1+x+x^2)^9 is?

Correct option is B. 84
Step 1 :- Simplify the expression
Given, (1+x)(1-x)^10(1+x+x^2)^9
(1+x)(1-x)^10(1+x+x^2)^9 = (1+x)(1-x)(1-x)^9(1+x+x^2)^9
We know that, (a^3-b^3)=(a-b)(a^2+ab+b^2)
(1+x)(1-x)^10(1+x+x^2)^9 = (1-x^2)(1-x^3)^9
Step 2 :- Find the coefficient of x^18
We know that
The (r+1)th term in the binomial expansion (a+b)^n is given by nCr a^(n-r) b^r
The (r+1)th term in the binomial expansion (1-x^3)^n = nCr (-x^3)^r = nCr (-1)^r x^(3r)
The coefficient of x^18 in the binomial expansion (1-x^2)(1-x^3)^9 =
The coefficient of x^18 in the binomial expansion (1-x^3)^9 = 9C6(-1)^6 x^18 = 84
The coefficient of x^16 in the binomial expansion (1-x^3)^9 does not exist. Because there is no r which is integer for 3r=16
From equation (3) The coefficient of x^18 in the binomial expansion (1-x^2)(1-x^3)^9 = 84
Hence, B is the correct option.