The coefficient of `x^r[0lt=rlt=(n-1)]` in the expansion of `(x+3)^(n-1)+(x+3)^(n-2)(x+2)+(x+3)^(n-3)(x+2)^2+....` `+(x+2)^(n-1)` is `a.^n C_r(3^r-2^n)` `b.^n C_r(3^(n-r)-2^(n-r))` `c.^n C_r(3^r+2^(n-r))` d. none of these

Options

A. `""^(n)C_(r) 2^(n-r)`
B. `""^(n)C_(r) (2^(n-r) -1)`
C. `""^(n)C_(r) (2^(n-r) +1)`
D. `""^(n)C_(r) (2^(r) -1)`

Answer

b

Explanation

We have,
`(x +2)^(n-1) + (x +2)^(n -2) (x +1) + (x +2)^(n-3) (x +1)^(2) +…+ (x +1) ^(n-1)`,
`(x +2)^(n-1){(1-((x+1)/(x+2))^(n))/(1-(x +1)/(x+2))} = (x +2)^(n) - (x+1)^(n)`
`therefore ` Coefficient of `x^(r)`in `(x +2)^(n)` - Coefficient of `x^(r)` in `(x + 1)^(n)`
`""^(n)C_(2) 2^(n-r) - ""^(n)C_(r) (2 ^(n-r) 1) ""^(n)C_(r)` .