First slide

Permutations

Question

If a represents the number of permutations of (x + 2) things taken together, b represents the number of permutations of 11 things taken together out of x things, and c represents the number of permutations of (x – 11) things taken together so that a = 182bc, then x =

Moderate

A

15
B

12
C

10
D

18

Solution

$\begin{array}{l} a =^{x + 2} P_{x + 2}, b =^{x} P_{11}, C =^{x - 11} P_{x - 11} \\ \Rightarrow a = (x + 2)!, b = \frac{x!}{(x - 11)!}, c = (x - 11)! \end{array}$
Now, a = 182 bc
$\begin{array}{l} \Rightarrow (x + 2)! = 182 \frac{x!}{(x - 11)!} (x - 11)! \\ \Rightarrow (x + 2) (x + 1) x! = 182 x! \\ \Rightarrow (x + 2) (x + 1) = 182 = 14 \times 13 \\ \Rightarrow x + 1 = 13 \\ ∴ x = 12 \end{array}$

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