Example 6 Given a positive integer $n$. How many ordered quadruples of integers $(a, b, c, d)$ satisfy the condition
$$
0 \leqslant a \leqslant b \leqslant c \leqslant d \leqslant n
$$

?
Define a mapping $f$:
$$
(a, b, c, d) \mapsto(a, b+1, c+2, d+3) .
$$

Then $f$ is a mapping from the set $A=\{(a, b, c, d) \mid 0 \leqslant a \leqslant b \leqslant c \leqslant d \leqslant n\}$ to the set $B=\{(a, b', c', d') \mid 0 \leqslant a < b' < c' < d' \leqslant n+3\}$. It is easy to verify that this mapping is a one-to-one correspondence. Therefore, $|A|=|B|$.

Since $|B|$ is the number of four-element subsets of the set $\{0,1,2, \cdots, n+3\}$, which is $\mathrm{C}_{n+4}^{4}$, it follows that $|A|=\mathrm{C}_{n+4}^{4}$.
Explanation Using one-to-one correspondence can help us solve many combinatorial counting problems. When we want to find the number of elements in set $A$, we can find a set $B$ that can establish a one-to-one correspondence with set $A$ and is easy to count, and then calculate the number of elements in set $B$. This example is exactly such a case. Below is another interesting problem.