【Example 3】 How many positive integers can 2160 be divided by? How many positive even integers can it be divided by?
Solve the prime factorization of 2160, we get
$$
2160=2^{4} \times 3^{3} \times 5
$$

Therefore, every positive divisor of it should have the form
$$
2^{i} \cdot 3^{j} \cdot 5^{k}
$$

where $i, j, k$ are integers, and $0 \leqslant i \leqslant 4, 0 \leqslant j \leqslant 3, 0 \leqslant k \leqslant 1$. So to determine a positive divisor of 2160, we need to select $i, j, k$ once within their respective ranges, which is a triple selection problem. By the multiplication principle, it has
$$
5 \times 4 \times 2=40
$$

different positive divisors. Since the positive even divisors of 2160 must contain the factor 2, so $1 \leqslant i \leqslant 4, 0 \leqslant j \leqslant 3, 0 \leqslant k \leqslant 1$, hence it has
$$
4 \times 4 \times 2=32
$$

positive even divisors.