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

As many as you'd like

Answer 2

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.