1.

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

Answer»

`.^(1001)C_(50)`
`.^(1002)C_(951)`
`.^(1001)C_(950)`
`.^(1002)C_(50)`

Solution :`.^(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)]`
`=` coefficientof `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)`


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