Question

If four times the number of permutations of $n$ things $3$ together is equal to five times the number of permutations of $n-1$ things $3$ together, find $n$.

Answer 1

Given : $4\times {}^n{P}_3=5\times {}^{n-1}{P}_3$
$⟹4\times \frac{n!}{(n-3)!}=5\times \frac{(n-1)!}{(n-4)!}$
$⟹4\times n=5\times (n-3)$
$⟹4n=5n-15$
$⟹n=15$