The number of 5-digit natural numbers, such that the product of their digits is 36, is ___…

Question

The number of 5-digit natural numbers, such that the product of their digits is 36, is \_\_\_\_\_\_\_.

Answer: 180

Accepted Answer

$36 = 2^2\times3^2$. Enumerate 5-digit combinations (digits 1--9) whose product is 36:

- $(1,2,2,3,3)$: $\dfrac{5!}{2!2!}=30$
- $(1,1,4,3,3)$: $\dfrac{5!}{2!2!}=30$
- $(1,1,9,2,2)$: $\dfrac{5!}{2!2!}=30$
- $(1,1,1,4,9)$: $\dfrac{5!}{3!}=20$
- $(1,1,2,3,6)$: $\dfrac{5!}{2!}=60$
- $(1,1,1,6,6)$: $\dfrac{5!}{3!2!}=10$

$$\text{Total} = 30+30+30+20+60+10 = 180$$

Solutions · 1