Question

The number of ways $1080$ can be expressed as a product of three natural numbers is

Answer 1

$1080=2^3\times 3^2\times 5$
No. of ways of distributing powers of $2$ over $3$ variables $={}^{3+3-1}{C}_{3-1}={}^5{C}_2=10$
No. of ways of distributing powers of $3$ over $3$ variables $={}^{3+3-1}{C}_{3-1}={}^5{C}_2=10$
No. of ways of distributing powers of $5$ over $3$ variables $={}^{1+3-1}{C}_{3-1}={}^3{C}_2=3$
Total number of ordered triplets$=10\times 10\times 5=300$
But, we have to find unordered triplets.
Now, ordered triplets of form $(a,a,a)=0$ as $a^3\ne 1080$
Ordered triplets of form $(a,a,b)$ will be the one where one of the factor of $1080$ is a perfect square i.e. $1,4,9,36$
Number of ordered pairs corresponding to them$=4\times 3=12$
Number of distinct triplets of form $(a,b,c)=288$
Therefore, number of unordered triplets$=\frac{288}{3!}+4=48+4=52$