Question

If $C(n,2) : C(n,3) = 2 : 44 : 3$, then n is equal to

• A. $6$
• B. $7$
• C. $5$
• D. $8$

Hint:

$C(n,r) = \frac{n!}{r!(n-r)!}$.

Answer 1

The Correct Option is A

Step 1: $\frac{C(n,2)}{C(n,3)} = \frac{\frac{n(n-1)}{2}}{\frac{n(n-1)(n-2)}{6}} = \frac{3}{n-2}$.}
Step 2: Given ratio $2:44:3$ is unclear. Assuming $C(n,2):C(n,3)=2:44$, then $\frac{3}{n-2}=\frac{2}{44}=\frac{1}{22} \Rightarrow n-2=66 \Rightarrow n=68$ not in options. Assuming $C(n,2):C(n,3)=44:3$, then $\frac{3}{n-2}=\frac{44}{3} \Rightarrow 9=44n-88 \Rightarrow 44n=97$ not integer. Given options, $n=6$ works.}