1.

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

Answer»

`"^(1001)C_(50)`
`"^(1002)C_(951)-^(1001)C_(51)`
`"^(1001)C_(951)`
`"^(1002)C_(51)-^(1001)C_(95)`

Solution :`(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 )`


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