Question

If the coefficient of pth term in the expansion of ${(1+x)}^{44}$ is $p$ and that of $(p+1)th$ term is $q$, then $(p+q-1)=$

• A. $n$
• B. $2n$
• C. $3n$
• D. $4n$

Answer 1

Answer: (A)

$n$

We have,

$(1+x{)}^n{=}^n{C}_0{+}^n{C}_{1}x{+}^n{C}_2{x}^2+.....{+}^n{C}_r{x}^r+......{+}^n{C}_n{x}^n$

$\therefore (1+x{)}^{44}{=}^{44}{C}_0{+}^{44}{C}_{1}x{+}^{44}{C}_2{x}^2+.....{+}^{44}{C}_{p-1}{x}^{p-1}{+}^{44}{C}_p{x}^p+......{+}^{44}{C}_{44}{x}^{44}$

Coefficient of $p^{th}$ term $=p$

i.e., ${}^{44}{C}_{p-1}=p$

$\Rightarrow p=1,p=44$

Coefficient of $(p+1{)}^{th}$ term $=q$

i.e., ${}^{44}{C}_p=q$

$\therefore p+q-1{=}^{44}{C}_{p-1}{+}^{44}{C}_p-1{=}^{45}{C}_p-1=44=n$