Question

If \( \frac{C(2n,3)}{C(n,2)} = \frac{44}{3} \), then \( n \) is equal to

• A. 6
• B. 7
• C. 5
• D. 8

Hint:

Cancel common factors like \((n-1)\) early to simplify quickly.

Answer 1

The Correct Option is A

Concept: \[ C(n,r)=\frac{n!}{r!(n-r)!} \]

Step 1: Expand.
\[ C(2n,3)=\frac{(2n)(2n-1)(2n-2)}{6},\quad C(n,2)=\frac{n(n-1)}{2} \]

Step 2: Form equation.
\[ \frac{(2n)(2n-1)(2n-2)}{6} \cdot \frac{2}{n(n-1)} = \frac{44}{3} \] \[ \Rightarrow \frac{(2n)(2n-1)(2n-2)}{3n(n-1)} = \frac{44}{3} \]

Step 3: Simplify.
\[ \frac{2(2n-1)(2n-2)}{(n-1)} = 44 \] Cancel \((n-1)\): \[ 2(2n-1)\cdot 2 = 44 \Rightarrow 4(2n-1)=44 \] \[ 2n-1=11 \Rightarrow n=6 \]