Question

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

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

Answer 1

The correct option is B $21$

Let the number of terms $n=2m$

According to question

Coefficient of largest term=${}^{2m}{C}_m$

Second largest coefficient tern =${}^{2m}{C}_{m-1}$

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

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

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

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

$⟹10m+10=11m⟹m=10$

$⟹n=2m=20$

So number of terms is 20.