Problem 614

1. The mapping $(i, j, k, h) \rightarrow(i, j, k+1, h+1)$ is a one-to-one mapping from the set $A=\{(i, j, k, h) \mid 1 \leqslant i<j \leqslant k<h \leqslant n+1\}$ to the set $B=\left\{\left(i, j, k^{\prime}, h^{\prime}\right) \mid 1 \leqslant i<j<k^{\prime}<h^{\prime} \leqslant n+2\right\}$, so $|A|=|B|$. And $|B|$ is clearly the number of four-element subsets of the set $\{1,2, \cdots, n+2\}$, which is $C_{n+2}^{4}$. Therefore, $|A|=C_{n+2}^{4}$.