The problem states that there are 10 students volunteering for community work. The community leader assigns 5 students to visit villages to advise people, 3 students to assist with administrative tasks, and 2 students to gather information. The question asks how the leader can divide the students according to these tasks. It seems like it is asking for the number of ways to divide the 10 students into three groups of sizes 5, 3, and 2.

1. Problem Description

The problem states that there are 10 students volunteering for community work. The community leader assigns 5 students to visit villages to advise people, 3 students to assist with administrative tasks, and 2 students to gather information. The question asks how the leader can divide the students according to these tasks. It seems like it is asking for the number of ways to divide the 10 students into three groups of sizes 5, 3, and

2.

2. Solution Steps

We need to find the number of ways to divide 10 students into three groups of 5, 3, and

2. This is a combination problem.

First, we choose 5 students out of 10 for the first task (advising). The number of ways to do this is given by the combination formula:

$C(n, k) = \frac{n!}{k!(n-k)!}$

$C(10, 5) = \frac{10!}{5!(10-5)!} = \frac{10!}{5!5!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252$

Next, we choose 3 students out of the remaining 5 for the second task (administration). The number of ways to do this is:

$C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = \frac{5 \times 4}{2 \times 1} = 10$

Finally, we choose 2 students out of the remaining 2 for the third task (information gathering). The number of ways to do this is:

$C(2, 2) = \frac{2!}{2!(2-2)!} = \frac{2!}{2!0!} = 1$

Now, we multiply these values together to find the total number of ways to divide the students:

Total ways = $C(10, 5) \times C(5, 3) \times C(2, 2) = 252 \times 10 \times 1 = 2520$

3. Final Answer

The community leader can divide the students in 2520 ways.