In a lottery draw of five numbers, when the winning numbers are written in ascending order, which number appears most frequently in the second position?
Solution. Let $m(i)$ denote the number of draws in which the number $i$ $(2 \leq i \leq 87)$ is the second in the sorted order; then

$$
m(i)=\binom{i-1}{1}\binom{90-i}{3}
$$

since one must be chosen from $i-1$ elements for the first position, and three from 90 - $i$ for the third, fourth, and fifth positions. We break down the binomial coefficients:

$$
m(i)=\binom{i-1}{1}\binom{90-i}{3}=\frac{(i-1)(90-i)(89-i)(88-i)}{6}
$$

This is a fourth-degree function defined on the set $\mathbb{Z} \cap[2 ; 87]$, which only takes positive values.

We examine where the function increases and decreases. The function increases if

$$
\begin{aligned}
m(i) &< m(i+1); similarly, this occurs when $i>22.5$. Therefore, the function decreases between 23 and 87.

This and the previous statement together imply that the sought value is 22 or 23.

$$
\begin{aligned}
& m(22)=\frac{(22-1)(90-22)(89-22)(88-22)}{6}=\frac{21 \cdot 68 \cdot 67 \cdot 66}{6}=\frac{6314616}{6} \\
& m(23)=\frac{(23-1)(90-23)(89-23)(88-23)}{6}=\frac{22 \cdot 67 \cdot 66 \cdot 65}{6}=\frac{6323460}{6}
\end{aligned}
$$

Since $m(23)>m(22)$, the number 23 appears most frequently in the second position when sorted by size.