4. In how many ways can the number 210 be factored into the product of four natural numbers? The order of the factors does not matter.

$(12$ points)
# Solution. 

Factorize 210 into a product of prime numbers: $2 \cdot 3 \cdot 5 \cdot 7$. Let's see how 4 prime


#### Abstract

divisors can be distributed among the desired factors


1) $4+0+0+0-1$ way.
2) $3+1+0+0-C_{4}^{3}=4$ ways.
3) $2+2+0+0-C_{4}^{2} / 2=3$ ways.
4) $2+1+1+0-C_{4}^{2}=6$ ways.
5) $1+1+1+1-1$ way.

Answer: 15.

## Answer: 15