A professor who teaches a number of courses is choosing a team of 4 boys and 3 girls from among his graduate students to help with the setup of the labs in his courses. He has 12 graduate students of 6 boys and 6 girls.
Assuming that the order of students does not matter, in how many different ways can he form a team of 4 boys and 3 girls from his graduate students?
To determine the number of different ways the professor can form a team of 4 boys and 3 girls from his graduate students, we can use the concept of combinations.
The number of ways to choose 4 boys from a group of 6 boys can be calculated using the formula for combinations, which is represented as:
C(n, r) = n! / (r!(n - r)!)
where n is the total number of objects, r is the number of objects to be chosen, and ! denotes the factorial function.
For our scenario, we have 6 boys and need to choose 4 of them.
C(6, 4) = 6! / (4!(6 - 4)!)
= 6! / (4!2!)
Similarly, the number of ways to choose 3 girls from a group of 6 girls can be calculated as:
C(6, 3) = 6! / (3!(6 - 3)!)
= 6! / (3!3!)
To find the total number of ways the professor can form a team of 4 boys and 3 girls, we need to multiply the two combinations together:
Total number of ways = C(6, 4) * C(6, 3)
= (6! / (4!2!)) * (6! / (3!3!))
Now, let's calculate this:
First, calculate C(6, 4):
C(6, 4) = 6! / (4!2!)
= (6 * 5 * 4!) / (4! * 2 * 1)
= (6 * 5) / (2 * 1)
= 15
Next, calculate C(6, 3):
C(6, 3) = 6! / (3!3!)
= (6 * 5 * 4!) / (3! * 3 * 2 * 1)
= (6 * 5) / (3 * 2 * 1)
= 10
Now, multiply both combinations together:
Total number of ways = 15 * 10
= 150
Therefore, there are 150 different ways the professor can form a team of 4 boys and 3 girls from his graduate students.