Question

How many different committees can be formed from 8 teachers and 48 students if the committee consists of 3 teachers and 4 ​students?

Answer 1

Since we don't care about the order in which we select people, we will use the combinatoric approach, rather than permutations.
Since we want to select 3 teachers from a total of 8, this is the notation to select 3 teachers from 8 (objects).

Teachers :
`^(8)C_3`
Similarly, students :
`^(48)C_(4)`

Now, since we want both at the same time, we will need to multiply both to yield your final answer:
`^(8)C_3 \cdot ^(48)C_(4) = 10,896,480` different ways.

Answer 2

As many as you'd like