Question

Find the coefficient of `x^(4)` in the expansion of `(1+x)^(n)(1-x)^(n).` Deduce that `C_(2) = C_(0)C_(4) - C_(1)C_(3) +C_(2)C_(2)-C_(3)C_(1)+ C_(4)C_(0)," where "C_(r) " stands for " ^(n)C_(r).`

Answer 1

Correct Answer - `C_(2)`
`(1+x)^(n) xx ( 1-x)^(n) = [C_(0) + C_(1)x + C_(2)x^(2) + ...+ C_(n) x^(n)] xx [C_(0) - C_(1)x + C_(2)x^(2) - ... + (-1).^(n)C_(n)x^(n)]`
`(1+x)^(n) xx (1-x)^(n) = (1-x^(2))^(n) = [C_(0)- C_(1)x^(2) + C_(2)x^(4)-...+(-1).^(n)C_(n)x^(2n)]`
`:. C_(0)C_(4) - C_(1) C_(3) + C_(2)C_(2)- C_(3) C_(1) + C_(4)C_(0) = C_(2).`