Indian Statistical Institute B.Math & B.Stat : Combinatorics

What is the number of ordered triplets $(a,b,c)$ where $a,b,c$ are positive integers ( not necessarily distinct ) such that $abc=1000.$ Since $1000=2^3 5^3$ any ordered triplet $(a,b,c)$ must be of the form $(2^i5^p,2^j5^q,2^k5^r)$ where $i+j+k=3$, $p+q+r=3$ and $i,j,k,p,q,r$ are non-negative integers. Number of solutions to the equation $i+j+k=3$ and $p+q+r=3$ is given by $\binom{3+3-1}{3-1} =10$. Now for each set of values of $i,j,k$ there are $10$ possible combinations of $p,q,r$. Thus giving a total of $10 \times 10 =100$ ordered triplets. Note the number of solutions to the equation $x_1+x_2+\dots+x_r=n, n \in \mathbb{N}$ in positive integers is $\binom {n-1}{r-1}$ and in non-negative integers is $\binom {n+r-1}{r-1}$.