Scummydale Senior High School has 8 math teachers and 4 history teachers. Suppose that a committee of 8 of those teachers is formed to attend a local conference.
Ok. So???
To find the number of ways a committee of 8 teachers can be formed from 8 math teachers and 4 history teachers, we can use the combination formula.
The combination formula is given by:
C(n, r) = n! / (r!(n-r)!)
Where:
- C(n, r) represents the number of combinations of n things taken r at a time,
- n! represents the factorial of n (n * (n-1) * (n-2) * ... * 2 * 1),
- r! represents the factorial of r,
- (n-r)! represents the factorial of (n-r).
In this case, we have 8 math teachers and 4 history teachers, so the total number of teachers to choose from is 8+4=12 (n=12). We want to choose a committee of 8 teachers (r=8).
Using the combination formula, we can calculate the number of ways the committee can be formed:
C(12, 8) = 12! / (8!(12-8)!)
= 12! / (8!4!)
Calculating the factorials:
12! = 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
4! = 4 * 3 * 2 * 1
Plugging in the values:
C(12, 8) = (12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) * (4 * 3 * 2 * 1))
Simplifying:
C(12, 8) = (12 * 11 * 10 * 9) / (4 * 3 * 2 * 1)
= 495
Therefore, there are 495 ways to form a committee of 8 teachers from the 8 math teachers and 4 history teachers at Scummydale Senior High School.