The value of `"^(1000)C_(50)+^(999)C_(49)+^(998)C_(48)+......+^(950)C_(0)` is

Options

A. `.^(1001)C_(50)`
B. `.^(1002)C_(951)`
C. `.^(1001)C_(950)`
D. `.^(1002)C_(50)`

Answer

A

Explanation

`.^(1000)C_(50)+.^(999)C_(49)+.^(998)C_(48)+"...."+.^(950)C_(0)`
`=` coefficient of `x^(950)` in `[(1+x)^(950) + (1+x)^(951) + "....."+(1+x)^(1000)]`
`=` coefficient of `x^(950)` in `((1+x)^(950)((1+x)^(51)-1))/((1+x)-1)`
`=` coefficient of `x^(950)` in `((1+x)^(1001) - (1+x)^(950))/(x)`
`=` coefficient of `x^(951)` in `(1+x)^(1001)`
`= .^(1001)C_(1951) = .^(1001)C_(50)`