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

Options

A. (a) `""^(1001)C_(50)`
B. (b) `""^(1002)C_(951)`
C. (c) `""^(1001)C_(950)`
D. (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)`