Calculating $\tau(n)$, the number of positive divisors of $n \in \mathbb N$

In (a) you can write each positive divisor uniquely in the form

$$p_1^{\ell_1}p_2^{\ell_2}\ldots p_k^{\ell_k}\;,$$

where $\ell_i\in\{0,1,\ldots,a_i\}$ for $i=1,\ldots,k$. Choosing a positive divisor amounts to choosing the exponents $\ell_1,\ldots,\ell_k$; in how many ways can that be done?

Your answer to (b) is way off, partly because you did not in fact apply the formula, and partly because you didn’t finish decomposing $1,633,500$ as a product of powers of distinct primes.

Your answer to (c) is fine.