Question

The coefficient of $x^r[0\le r\le (n-1\left)\right]$ in the expansion of ${(x+3)}^{n-1}+{(x+3)}^{n-2}(x+2)+{(x+3)}^{n-3}{(x+2)}^2+.....+{(x+2)}^{n-1}$ are

• 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 1

Answer: (A)

${}^n{C}_r(3^r-2^n)$

We know that,

$\frac{x^n-a^n}{x-a}=x^{n-1}+x^{n-2}{a}^1+x^{n-3}{a}^2+....+a^{n-1}$

$\therefore (x+3{)}^{n-1}+(x+3{)}^{n-2}(x+2)+(x+3{)}^{n-3}(x+2{)}^2+...+(x+2{)}^{n-1}$

$=\frac{(x+3{)}^n-(x+2{)}^n}{(x+3)-(x+2)}=(x+3{)}^n-(x+2{)}^n$

$\therefore$ Coefficient of $x^r$ in the given expression $=$ Coefficient of $x^r$ in $\left[\right(x+3{)}^n-(x+2{)}^n]$

$={}^n{C}_r{3}^{n-r}-{}^n{C}_r{2}^{n-r}={}^n{C}_r(3^{n-r}-2^{n-r})$

Hence, the correct option is (A).