Q199 / 270 · NDA 2020 · Mathematics

If $C(20, n+2)=C(20, n-2)$, then what is $n$ equal to?

  • A
    18
  • B
    25
  • Correct answer
    C
    10
  • D
    12

Solution

${ }^{20} C_{(n+2)}={ }^{20} C_{(n-2)}$ $\Rightarrow \frac{20!}{(n+2)!(20-n-2)!}=\frac{20!}{(n-2)!(20-n+2)!}$ $\begin{aligned} & \Rightarrow \frac{(22-n)!}{(18-n)!}=\frac{(n+2)!}{(n-2)!} \\ & \Rightarrow(22-n)(21-n)(20-n)(19-n) \\ & \quad=(n+2)(n+1) \cdot n \cdot(n-1)\end{aligned}$ For $n=10$ $\begin{aligned} & (22-10)(21-10)(20-10)(19-10) \\ & =(10+2)(10+1) \cdot 10 \cdot(10-1) \\ & \Rightarrow 12 \cdot 11 \cdot 10 \cdot 9=12 \cdot 11 \cdot 10 \cdot 9 \end{aligned}$ Hence, $n=10$.

Hence, the correct answer is option: 'C'.

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