Question

A nine digit number is formed from $1,2,3,4,5$ such that product of all digits is always $1920$. The total number of ways is $393{(}^9{P}_r)$, where the value of $r$ is

Answer 1

1920

Prime factorization of $1920=2^7\times 3\times 5$

Possible ways

$2^7\times 3\times 5=\frac{9!}{7!}=72$

$4\times 2^5\times 3\times 5\times 1=\frac{9!}{5!}=3024$

$4\times 4\times 2^3\times 3\times 5\times 1^2=\frac{9!}{3!2!2!}=15120$

$4\times 4\times 4\times 2\times 3\times 5\times 1\times 1\times 1=\frac{9!}{3!3!}=10080$

$393\left({}^9{P}_r\right)=72+3024+15120+10080$

$393\left({}^9{P}_r\right)=28296$

${}^9{P}_r=\frac{28296}{393}$

${}^9{P}_r=72$

$\frac{9!}{(9-r)!}=72$

$\frac{9!}{(9-r)!}=\frac{9!}{7!}$

$9-r=7$

$r=2$