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Definition of Function

If p=n+2pn+...

Question

If $p =^{n + 2} p_{n + 2}$ ; $q =^{n} p_{11}$ ; $r =^{n - 11} p_{n - 11}$ and if p =182 qr, then show that the value of r is 12.

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Solution

p = (n+2)!, $q = \frac{n!}{(n - 11)!}$ ; r = (n-11)!
from the given relation we have
$(n + 2)! = 182 \frac{n!}{(n - 11)!} (n - 11)! = 182 n!$
(n+2)(n+1) = 182
$n^{2} + 3 n - 180 = 0$
or (n+15)(n-12)=0
n=12(+ve value)

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