The coefficient of x10in the expansion of 1+x2−x38is

We rewrite the given expression as 1+x2(1−x)8and expand by using the binomial theorem. We have, 1+x2(1−x)8=8C0+8C1x2(1−x)+8C2x4(1−x)2+8C3x6(1−x)3+8C4x8(1−x)4+8C5x10(1−x)5+…The two terms which contain x10 are 8C4x8(1−x)8 and 8C5x10(1−x)5.Thus, the coefficient of x10 in the given expression is given by 8C4[coefficient of x2 in the expansion of (1−x)4+8C3=8C4(6)+8C5=8!4!4!(6)+8!3!5!=(70)(6)+56=476