How many ways can 15000 be written as the product of 3 integers?

Question

I've looked at questions similar to these, such as (In how many possible ways can we write $3240$ as a product of $3$ positive integers $a$, $b$ and $c$?). I don't understand how he got 10 when he split it among 3 divisors.

The prime factorization of $15000$ is $2^3 \cdot 3 \cdot 5^4$. Can anyone use this to explain to me?

Answer 1

You know how to place eggs in baskets with the bar method?

https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)

1. Divide 3 eggs labeled '2' over 3 baskets in $5\choose{2}$ ways by placing 2 bars in 3+2 positions.
2. Divide 1 egg labeled '3' over 3 baskets, 3 ways
3. Divide 4 eggs labeled '5' over 3 baskets. $6 \choose 2$

Multiply..