2. (10 points) Minister K. issued an order that citizens will only be received if the number of ways to choose a group of four people from those who have come is less than the number of ways to choose a group of two people from them. Determine the maximum number of citizens that the minister can receive?
Answer: 5.

Solution: The number of ways to choose a group of four people from $n$ people is $C_{n}^{4}$; a group of two people is $C_{n}^{2}$. We are interested in the maximum natural number $n$ such that $C_{n}^{4}<C_{n}^{2}$. Transforming:

$$
\frac{n(n-1)(n-2)(n-3)}{1 \cdot 2 \cdot 3 \cdot 4}<\frac{n(n-1)}{2} \Leftrightarrow \frac{(n-2)(n-3)}{3 \cdot 4}<1
$$

Solving the inequality using the interval method, we get that $-1<n<6$. Therefore, the maximum $n$ is 5.