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

Options

A. `"^(1001)C_(50)`
B. `"^(1002)C_(951)-^(1001)C_(51)`
C. `"^(1001)C_(951)`
D. `"^(1002)C_(51)-^(1001)C_(95)`

Answer

A, B, C, D

Explanation

`(a,b,c,d)` `'^(1000)C_(50)+^(999)C_(40)+^(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+(1+x)+(1+x)^(2)+….+(1+x)^(50)}`
`="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)`
`=^(1000)C_(951)=^(1001)C_(50)=^(1002)C_(951)-^(1001)C_(51)=^(1002)C_(51)-^(1001)C_(950)`, Since `'^(n)C_(r )+^(n)C_(r-1)=^((n+1))C_(r )`