Find the coefficient of `x^5` in the expansion of `(1+x)^(21)+(1+x)^(22)++(1+x)^(30)dot`

Options

A. `""^(51)C_(5)`
B. `""^(9)C_(5)`
C. `""^(31)C_(6)-""^(21)C_(6)`
D. `""^(30)C_(5)-""^(20)C_(5)`

Answer

C

Explanation

We have,
`(1 + x )^(21) + (1 + x)^(22) + … + (1 + x)^(30)`
`(1 + x )^(21) +{((1+ x)^(10) - 1)/((1+ x) -1)}= (1)/(x){ (1 + x)^(31) - (1 + x)^(21)}`
`therefore ` Coff. Of `x^(5)` in the given expanssion
= Coeff. Of `x^(5)` in ` [ (1)/(x) {(1 + x)^(31) - ( 1+ x)^(21)}`
= Coeff. of `x^(6)` in `{ (1 + x)^(31) - (1+ x)^(21)}`
`""^(31)C_(6) - ""^(21)C_(6)` .