If the coefficients of \(x^9\) and \(x^{10}\) in the binomial expansion of \(\left(3+\frac{x}{2}\right)^n\)…
If the coefficients of \(x^9\) and \(x^{10}\) in the binomial expansion of \(\left(3+\frac{x}{2}\right)^n\) are equal, then \(n=\)
69
96
66
99
Solution
Given binomial is \(\left(3+\frac{x}{2}\right)^n\) and the general term in the expansion is
\({ }^n C_r \frac{3^{n-r}}{2^r} x^r\)
\(\therefore\) Coefficient of \(x^9\) and \(x^{10}\) are
\({ }^n C_9 \frac{3^{n-9}}{2^9} \text { and }{ }^n C_{10} \frac{3^{n-10}}{2^{10}} \text { respectively }\)
\(\because\) It is given that, the coefficients \(x^9\) and \(x^{10}\) are equal,
\(\begin{aligned}
& \text {So, } { }^n C_9 \frac{3^{n-9}}{2^9} ={ }^n C_{10} \frac{3^{n-10}}{2^{10}} \\
& \Rightarrow \frac{n ! \times 3}{9 !(n-9) !} =\frac{n !}{10 !(n-10) ! \times 2} \\
\Rightarrow & \frac{3}{n-9} =\frac{1}{10 \times 2} \Rightarrow n-9=60 \\
\Rightarrow & n =69
\end{aligned}\)
Hence, option (a) is correct.