Problem Archive

Let $W(n,k)$ be the number of ways in which $n$ can be written as the product of $k$ distinct positive integers.

For example, $W(144,4) = 7$. There are $7$ ways in which $144$ can be written as a product of $4$ distinct positive integers:

• $144 = 1 \times 2 \times 4 \times 18$
• $144 = 1 \times 2 \times 8 \times 9$
• $144 = 1 \times 2 \times 3 \times 24$
• $144 = 1 \times 2 \times 6 \times 12$
• $144 = 1 \times 3 \times 4 \times 12$
• $144 = 1 \times 3 \times 6 \times 8$
• $144 = 2 \times 3 \times 4 \times 6$

Note that permutations of the integers themselves are not considered distinct.

Furthermore, $W(100!,10)$ modulo $1\,000\,000\,007 = 287549200$.

Find $W(10000!,30)$ modulo $1\,000\,000\,007$.