First slide

Permutations

Question

If $^{n + 2} C_{3} =^{n + 3} P_{2} - 20$ then $n$ is equal to:

Moderate

A

6
B

5
C

4
D

3

Solution

$\frac{1}{6} (n + 2) (n + 1) n = (n + 3) (n + 2) - 20$
$\begin{array}{ll} \Rightarrow n^{3} + 3 n^{2} + 2 n = 6 (n^{2} + 5 n + 6) - 120 \\ \Rightarrow n^{3} - 3 n^{2} - 28 n + 84 = 0 \\ \Rightarrow (n - 3) (n^{2} - 28) = 0 \end{array}$
As $n$ is a positive integer we get $n = 3$

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