Example 11 Given a positive integer $n$, find the number of ordered quadruples of integers $(a, b, c, d)$ (where $0 \leqslant a \leqslant b \leqslant c \leqslant d \leqslant n$).
Consider the mapping
$$
f:(a, b, c, d) \mapsto(a, b+1, c+2, d+3),
$$

then this mapping $f$ is a one-to-one correspondence (bijection) from the set
$$
X=\{(a, b, c, d) \mid 0 \leqslant a \leqslant b \leqslant c \leqslant d \leqslant n\}
$$

to the set
$$
Y=\left\{\left(a^{\prime}, b^{\prime}, c^{\prime}, d^{\prime}\right) \mid 0 \leqslant a^{\prime}<b^{\prime}<c^{\prime}<d^{\prime} \leqslant n+3\right\}
$$

Therefore, the number of elements in set $X$ is equal to the number of elements in set $Y$.
The number of elements in set $Y$ is the number of 4-element subsets of the set $\{0,1,2, \cdots, n+3\}$, which is $C_{n+4}^{4}$. Thus, the number of ordered quadruples $(a, b, c, d)$ is $\mathrm{C}_{n+4}^{4}$.