how many different committees can be formed from 6 teachers and 31 students if the committee consists of 4 teachers and 4 students?
C(6,4) * C(31,4) = 15*31,465 = 471,975
To find the number of different committees that can be formed from 6 teachers and 31 students, where each committee consists of 4 teachers and 4 students, we can follow these steps:
Step 1: Determine the number of ways to select 4 teachers from the available 6 teachers.
This can be calculated using the combination formula: C(n, r) = n! / (r!(n-r)!).
In this case, n = 6 (total number of teachers) and r = 4 (number of teachers to be selected).
So, C(6, 4) = 6! / (4!(6-4)!) = 6! / (4!2!) = (6 * 5 * 4 * 3) / (4 * 3 * 2 * 1) = 15.
Step 2: Determine the number of ways to select 4 students from the available 31 students.
Using the same combination formula: C(n, r) = n! / (r!(n-r)!).
In this case, n = 31 (total number of students) and r = 4 (number of students to be selected).
So, C(31, 4) = 31! / (4!(31-4)!) = 31! / (4!27!) = (31 * 30 * 29 * 28) / (4 * 3 * 2 * 1) = 27,405.
Step 3: Find the total number of committees by multiplying the results from step 1 and step 2.
Total number of committees = number of ways to select teachers * number of ways to select students = 15 * 27,405 = 411,075.
Therefore, there can be 411,075 different committees formed from 6 teachers and 31 students where each committee consists of 4 teachers and 4 students.