7・62. From $\{1,2, \cdots, n\}$, select triples $(a, b, c), a<b<c$ and for any two selected triples $(a, b, c)$ and $\left(a^{\prime}, b^{\prime}, c^{\prime}\right)$, the equation $a=a^{\prime}, b=$ $b^{\prime}, c=c^{\prime}$ holds for at most 1. What is the maximum number of such triples that can be selected?
[Solution] Let $H$ be the set of triples that satisfy the condition. For each $1<k<n$, let $H_{k}$ be the subset of $H$ where the middle number of each triple is $k$, i.e., $b=k$. Clearly,
$$
\left|H_{k}\right| \leqslant \min \{k-1, n-k\} \text {. }
$$

Summing up, we get
$$
\begin{aligned}
|H| & =\sum_{k=2}^{n-1}\left|H_{k}\right| \leqslant \sum_{k=2}^{n-1} \min \{k-1, n-k\} \\
& =\left\{\begin{array}{l}
\frac{n(n-2)}{4}, n \text { is even, } \\
\left(\frac{n-1}{2}\right)^{2}, n \text { is odd. }
\end{array}\right.
\end{aligned}
$$

When taking all the arithmetic triples from $\{1,2, \cdots, n\}$, the number is exactly the number on the right side of the above expression. Therefore, it is the maximum number we are looking for.