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Coefficient of x^m in (1 +x)^m+ (1+ x)^(...

Coefficient of `x^m` in `(1 +x)^m+ (1+ x)^(m+1)+........ (1+ x)^n, m < n` is

A

` ""^(n+1)C_(m+1)`

B

`""^(n-1)C_(m-1)`

C

`""^(n)C_(m)`

D

`""^(n)C_(m+1)`

Text Solution

Verified by Experts

The correct Answer is:
a
`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)`
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