Question

If the coefficients of $x^3$ and $x^4$ in the expansion of ${(3+kx)}^9$ are equal, then the value of $k$ is

• A. $3$
• B. $1/3$
• C. $2$
• D. $1/2$
• E. $1$

Answer 1

The correct option is C $2$

In expansion ${(3+kx)}^9$,
coefficients of $x^3$ and $x^4$ are equal
$\therefore {{}^{9}C}_3{\left(3\right)}^6{\left(k\right)}^3={{}^{9}C}_4{\left(3\right)}^5{\left(k\right)}^4[\because T_{r+1}={{}^{n}C}_r{a}^{n-r}{b}^r]$
$\Rightarrow \frac{9!}{3!6!}\left(3\right)=\frac{9!}{4!5!}\times k$
$\Rightarrow k=\frac{3\times 4\times 3!\times 5!}{3!\times 6\times 5!}$
$\therefore k=\frac{12}{6}=2$