Question

The expansion of ${\left(1+x\right)}^n$ has 3 consecutive terms with coefficient in the ratio $1:2:3$ and can be written in the form ${}^n{C}_k{;}^n{C}_{k+1}{:}^n{C}_{k+2}.$. The sum of all possible values of $n+k$ is

• A. 18
• B. 21
• C. 28
• D. 32

Answer 1

The correct option is A 18

Given

${}^n{C}_k{:}^n{C}_{k+1}{:}^n{C}_{k+2}=1:2:3$

$\frac{{}^n{C}_k}{{}^n{C}_{k+1}}=\frac{1}{2}\Rightarrow \frac{n!(n-k-1)}{(n-k)!x\left(k\right)!}\times \frac{(k+1)!}{n!}=dfrac12$

$\Rightarrow \frac{(k+1)}{(n-k)}=\frac{1}{2}$

$\Rightarrow 2k+2=n-k$

$\Rightarrow n=3k+2$------- $\left(1\right)$

$\frac{{}^n{C}_{k+1}}{{}^n{C}_{k+2}}=\frac{2}{3}\Rightarrow \frac{n!(n-k-2)!}{(n-k-1)!(k+1)!}\Rightarrow \frac{2}{3}$

$\Rightarrow \frac{(k+2)}{(n-k-1)}=\frac{2}{3}$

$\Rightarrow 3k+6=2n-2k-2$

$\Rightarrow 5k+8=2n$

On solving $6k+9=5k+8\Rightarrow k=4\Rightarrow n=x4+2$

$\therefore n+k=14+4=18$

$n=14$