Digit Product of 10000 (from Brilliant.org) How many 7 digit positive integers are there such that the product of the individual digits of each number is equal to 10000?
Details and assumptions
As an example, 22225555 is an 8 digit integer, whose product of individual digits is 10000.
Note: This is a past question from brilliant.org.

$10000 = 10^4 = 2^45^4$. The only way you can get the $5^4$ is by actually having $4$ copies of $5$. Now you need $2^4$, which can be divided to $(4,2,2)$ or $(4,4,1)$ or $(8,2,1)$.
If you select either of $(4,2,2)$ or $(4,4,1)$, the problem is creating a number from seven digits comprised of one set of $4$ identical ones, one set of $2$ identical ones, and another one. The number of ways to do that is $\frac{7!}{4!\cdot 2!}$ (set the order for all of them, then disregard the orders of each repeating digit).
If you select $(8,2,1)$, you get $\frac{7!}{4!}$.
In total, since those are all distinct (they are made of different digits) there are $2\cdot\frac{7!}{4! \cdot 2!} + \frac{7!}{4!} = 2\cdot\frac{7!}{4!} = 420$ numbers.
