The coefficient of x18 in the product (1+x)(1−x)101+x+x29is

Given expression is (1+x)(1−x)101+x+x29=(1+x)(1−x)(1−x)1+x+x29=1−x21−x39Now, coefficient of x18 in the product (1+x)(1−x)101+x+x29=coefficient of y18 in the product 1−x21−x39=coefficient of y18 in (1 - x3 )9- coefficient of x16in (1 - x3)9Since, (r + 1)th term in the expansion of1−x39 is 9Cr−x3r=9Cr(−1)rx3rNow, for x18, 3r = 18⇒r = 6and for x18 , 3r=18⇒r=6and for x16,3r=16⇒r=163∉N.'. Required coefficient is  9C6=9!6!3!=9×8×73×2=84