Find n if ^n C_ n-3 =84 - Vidyadip

Question

Find n if ${ }^n C_{n-3}=84$

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Answer

$\begin{array}{ll} & { }^n C_{n-3}=84 \\ \therefore \quad & \frac{n !}{(n-3) ![n-(n-3)] !}=84 \\ \therefore \quad & \frac{n(n-1)(n-2)(n-3) !}{(n-3) ! \times 3 !}=84 \\ \therefore \quad & n(n-1)(n-2)=84 \times 6 \\ \therefore \quad & n(n-1)(n-2)=9 \times 8 \times 7\end{array}$

Comparing on both sides, we get

$n =9$

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