How many different committees can be formed from 5 professors, and 15 students. If each committee is made up of 2 professors, and 10 students.

5 choose 2 * 15 choose 12

=5C2*15C12
=5!/(2!3!) * 15!/(12!3!)
=5*4/2! * 15*14*13/6
=10*455
=4550

To determine the number of different committees that can be formed, we need to calculate the combination or the number of ways to select 2 professors from a group of 5 and the number of ways to select 10 students from a group of 15.

Let's start with selecting 2 professors from a group of 5. We can use the combination formula, which is represented as nCr, where n is the total number of professors and r is the number of professors to be selected. The formula for combination is given by:

nCr = n! / (r! * (n-r)!)

In our case, n = 5 (total number of professors) and r = 2 (professors to be selected). Applying the combination formula:

5C2 = 5! / (2! * (5-2)!)
= 5! / (2! * 3!)
= (5 * 4 * 3!) / (2! * 3!)
= (5 * 4) / (2 * 1)
= 10

So, there are 10 different ways to select 2 professors from a group of 5.

Now, let's move on to selecting 10 students from a group of 15. Using the same combination formula:

15C10 = 15! / (10! * (15-10)!)
= 15! / (10! * 5!)
= (15 * 14 * 13 * 12 * 11 * 10!) / (10! * 5!)
= (15 * 14 * 13 * 12 * 11) / (5 * 4 * 3 * 2 * 1)
= 3003

So, there are 3003 different ways to select 10 students from a group of 15.

Now, to determine the total number of different committees that can be formed, we multiply the number of ways to select 2 professors by the number of ways to select 10 students:

Total number of committees = 10 * 3003
= 30,030.

Therefore, there are 30,030 different committees that can be formed.