From a group of 5 male students and 8 female students who have good performance in writing poems, a teacher selects 3 male students and 4 female students to present their work in a specific order. How many ways can the teacher select these students?

1. Problem Description

From a group of 5 male students and 8 female students who have good performance in writing poems, a teacher selects 3 male students and 4 female students to present their work in a specific order. How many ways can the teacher select these students?

2. Solution Steps

We need to find the number of ways to select 3 male students out of 5 and 4 female students out of 8, and then arrange the selected students in order.

First, we calculate the number of ways to choose 3 male students out of 5, which is given by the combination formula:

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

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

Next, we calculate the number of ways to choose 4 female students out of 8:

$C(8, 4) = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!} = \frac{8 \times 7 \times 6 \times 5 \times 4!}{4! \times 4 \times 3 \times 2 \times 1} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70$

Now, we have 3 male students and 4 female students, so we have a total of 7 students. The number of ways to arrange these 7 students in a specific order is given by the permutation formula:

$P(n) = n!$

$P(7) = 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040$

The total number of ways to select the students and arrange them is the product of the number of ways to choose the male students, the number of ways to choose the female students, and the number of ways to arrange the selected students:

Total ways = $C(5, 3) \times C(8, 4) \times 7! = 10 \times 70 \times 5040 = 700 \times 5040 = 3528000$

3. Final Answer

3528000