Find n if ^n C_ n-2 =15 - Vidyadip

Question

Find $n$ if ${ }^n C_{n-2}=15$

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Answer

$
\begin{aligned}
& { }^n C_{n-2}=15 \\
& \therefore{ }^n C_2=15 \ldots \ldots\left[\because{ }^n C_r={ }^n C_{n-r}\right] \\
& \therefore \frac{n !}{(n-2) ! 2 !}=15 \\
& \therefore \frac{n(n-1)(n-2) !}{(n-2) ! 2 \times 1}=15 \\
& \therefore n(n-1)=30 \\
& \therefore n(n-1)=6 \times 5
\end{aligned}
$
Comparing both sides, we get
$
\therefore \mathrm{n}=6
$

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