A committee of 3 teachers and 3 students is to be formed to judge a contest. If there are 7 students and 5 teachers to choose from, how many different committees could be formed?
7C3 * 5C3
To find the number of different committees that could be formed, we need to use the combination formula.
The combination formula is given by:
C(n, r) = n! / (r!(n-r)!)
Where n is the total number of options and r is the number of options to choose.
In this case, we have 7 students to choose from to form a committee of 3 students. Therefore, n = 7 and r = 3.
Using the combination formula, we can calculate:
C(7, 3) = 7! / (3!(7-3)!)
Calculating the factorial terms:
7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040
3! = 3 x 2 x 1 = 6
(7-3)! = 4! = 4 x 3 x 2 x 1 = 24
Substituting these values in the combination formula:
C(7, 3) = 5040 / (6 x 24) = 5040 / 144 = 35
Therefore, there are 35 different committees that could be formed.
To find the number of different committees that could be formed, we need to use the concept of combinations.
A combination is a way of selecting items from a larger set where the order does not matter. In this case, we need to select 3 teachers from a group of 5 and 3 students from a group of 7.
The number of different committees that could be formed can be found by calculating the combination of selecting 3 teachers from a group of 5 and 3 students from a group of 7, and then multiplying these two combinations together.
The formula for calculating a combination is:
nCr = n! / (r!(n-r)!)
Where n is the total number of items and r is the number of items to be selected.
For the teachers, we have n = 5 (total number of available teachers) and r = 3 (number of teachers to be selected). Thus, the combination for selecting 3 teachers from a group of 5 can be calculated as:
5C3 = 5! / (3!(5-3)!) = 5! / (3!2!) = (5 * 4 * 3!) / (3! * 2 * 1) = 10
Similarly, for the students, we have n = 7 (total number of available students) and r = 3 (number of students to be selected). Thus, the combination for selecting 3 students from a group of 7 can be calculated as:
7C3 = 7! / (3!(7-3)!) = 7! / (3!4!) = (7 * 6 * 5!) / (3! * 4 * 3 * 2 * 1) = 35
Now, we can multiply these two combinations together to get the total number of different committees that could be formed:
10 * 35 = 350
Therefore, there are a total of 350 different committees that could be formed.