Question

Let $n$ be a positive even integer. If the ratio of the largest coefficient and the $2^{nd}$ largest coefficient in the expansion of ${(1+x)}^n$ is $11:10$. Then, the number of terms in the expansion of ${(1+x)}^n$ is

• A. $20$
• B. $21$
• C. $10$
• D. $11$

Answer 1

Answer: (B)

$21$

Let the number of terms $n=2m$

Now by condition $\frac{\text{Largest coefficient in}{(1+x)}^n}{\text{Second largest coefficient in}{(1+x)}^n}=\frac{11}{10}$

$\Rightarrow \frac{{{}^{2m}C}_m}{{{}^{2m}C}_{m-1}}=\frac{11}{10}$

$\Rightarrow \frac{(m-1)!(m+1)!}{m!m!}=\frac{11}{10}$

$\Rightarrow \frac{1}{m}\frac{m(m-1)!(m+1)m!}{m!m!}=\frac{11}{10}$

$\Rightarrow \frac{m+1}{m}\frac{m!m!}{m!m!}=\frac{11}{10}$

$\Rightarrow 10m+10=11m\Rightarrow m=10$

$\therefore n=20$

Hence total number of terms

$=n+1=20+1=21$