Perfect Factor

For simplicity, the number in question, $2^{15} \times 3^{20} \times 5^{7}$ , will referred to as $x$ from now on.

Each square $s$ such that $s | x$ is of the form $2^{2m} \times 3^{2n} \times 5^{2p}$ where $0 \leq m \leq 7$ , $0 \leq n \leq 10$ , and $0 \leq p \leq 3$ . This means that for each variable $m$ , $n$ , and $p$ we have $8$ , $11$ , and $4$ options, respectively.

Thus, the number of perfect squares that divide $x$ is $8 \times 11 \times 4 = \boxed{352}$ .