1.

Find the coefficient of `x^r` in the expansion of `1 +(1 + x) + (1 + x)^2 +.......+ (1 + x)^n`.A. `""^(n)C_(r)`B. `""^(n+1)C_(r )`C. `""^(n+1)C_(r+1)`D. none of these

Answer» Correct Answer - c
We have,
`1 + (1 + x)+(1+x)^(2) +...+ (1 +x)^(n) = (1 - (1 +x)^(n+1))/(1-(1 +x))`
`rArr 1 = - (1)/(x) { 1 - (1 - x)^(n+1)}`
`rArr 1 = (1)/(x) (1 + x)^(n+1) - (1)/(x)`
`therefore ` Coefficient of `x^(r)` in `{ 1 + (1 + x) + (1 + x)^(2) +... + (1 + x)^(n)}`
= Coefficient of `x^(r)` in `{( (1 +x)^(n+1))/(x)- (1)/(x)}`
` = ""^(n-1)C_(r+1!)`


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