Questions: 8 students go in a group to a basketball game at their college. 4 of the students are wearing orange shirts and 4 of the students are wearing blue shirts. In how many ways can these students line up if they want to alternate shirt colors, starting with orange?

Solution Steps

To solve this problem, we need to consider the constraints given: 4 students are wearing orange shirts and 4 are wearing blue shirts, and they need to alternate shirt colors starting with orange.

1. First, we need to determine the number of ways to arrange the 4 students wearing orange shirts.
2. Next, we need to determine the number of ways to arrange the 4 students wearing blue shirts.
3. Since the students need to alternate starting with orange, we can multiply the number of ways to arrange the orange shirts by the number of ways to arrange the blue shirts to get the total number of ways to line up the students.

The number of ways to arrange the 4 students wearing orange shirts is given by the factorial of the number of students:

\[ \text{ways\_orange} = 4! = 24 \]

Similarly, the number of ways to arrange the 4 students wearing blue shirts is also given by the factorial of the number of students:

\[ \text{ways\_blue} = 4! = 24 \]

Since the students must alternate shirt colors starting with orange, the total number of ways to line up the students is the product of the arrangements of orange and blue shirts:

\[ \text{total\_ways} = \text{ways\_orange} \times \text{ways\_blue} = 24 \times 24 = 576 \]

Final Answer