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)(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

Answer

B

Explanation

We have
`(x+3)^(n-1)+(x+3)^(n-2)(x+2)+(x+3)^(n-3)(x+2)^(2)+"....."+(x+2)^(n-1)`
`=((x+3)^(n)-(x+2)^(n))/((x+3)-(x+2))=(x+3)^(n)-(x+2)^(n)`
`( :'(x^(n)-a^(n))/(x-a)=x^(n-1)+x^(n-2)a^(1)+x^(n-3)a^(2)+"....."+a^(n-1))`
Therefore, coefficient of `x^(r)` in the given expression is equal to coefficient of `x^(r)` in `[(x+3)^(n)-(x+2)^(n)]`, which is given by
`.^(n)C_(r)3^(n-r)-.^(n)C_(r)2^(n-r)=.^(n)C_(r)(3^(n-r)-2^(n-r))`