The value of `.^(n)C_(0) xx .^(2n)C_(r) - .^(n)C_(1)xx.^(2n-2)C_(r)+.^(n)C_(2)xx.^(2n-4)C_(r)+"…."` is equal to

Options

A. `.^(n)C_(r-n) xx 2^(2n-r)` if `r ge n`
B. 0, if `r lt n`
C. `.^(n)C_(r-n) xx 2^(n-r)` if `r ge n`
D. `.^(-n)C_(r-n) xx 2^(2n-r)` if `r lt n`

Answer

A, B

Explanation

`.^(n)C_(0)xx.^(2n-r)C_(r)-.^(n)C_(1) xx .^(2n-2)C_(r)+.^(n)C_(2)xx.^(2n-4)C_(r)+"...."`
= Coefficient of `x^(r)` in `(.^(n)C_(0) (1+x)^(2n)-.^(n)C_(1) (1+x)^(2n-2) + .^(n)C_(2)(1+x)^(2n-4)+"…..")`
= Coefficient of `x^(r)` in `(.^(n)C_(0)((1+x)^(2))^(n) - .^(n)C_(1)((1+x)^(2))^(n-1)+.^(n)C_(2)((1+x)^(2))^(n-2)+"....")`
`=` Coefficient of `x^(r)` in `((1+x)^(2)-1)^(n)`
= Coefficient of `x^(r)` in `(2x+x^(2))^(n)`
General term, `T_(p+1) = .^(n)C_(p)(2x)^(n-p)(x^(2))^(p) = .^(n)C_(p)2^(n-p)x^(n+p)`
For `x^(r ), n+p=r`, So, `p = r-n`.
`:.` Coefficient of `x^(r) = .^(n)C_(r-n) xx 2^(2n-r)`
This is non-zero if `r ge n` and zero if `r lt n`.