The coefficient of `x^m` in `(1+x)^m +(1+m)^(m+1) +...+(1+x)^n ,m≤n` is

Options

A. ` ""^(n+1)C_(m+1)`
B. `""^(n-1)C_(m-1)`
C. `""^(n)C_(m)`
D. `""^(n)C_(m+1)`

Answer

a

Explanation

`rArr (logx + 4 )(log x - 1) = 0`
` rArr log x = - 4,1`
` therefore x = 10^(-4),10`
` = ((1 + x)^(m){(1 + x)^(n-m+1)-1})/((1 + x) -1) = ((1+ x)^(n+1) - (1 + x)^(m))/(x)`
` therefore ` Coefficient of ` x^(m) ` in .
`(1 + x)^(m) + (1 + x)^(m+1) + ...+ (1 + x)^(n)`
or coefficient of ` x^(m) "in" ((1 + x)^(n+1) -(1 +x)^(m))/(x)`
or coefficient of ` x^(m+1) " in " (1 + x)^(n+1) -(1 +x)^(m)`
` ""^(n+1)C_(m+1) - 0 = ""^(n+ 1)C_(m+1)`