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 : [tex]^{8}C_3[/tex]
Similarly, students : [tex]^{48}C_{4}[/tex]
Now, since we want both at the same time, we will need to multiply both to yield your final answer: [tex]^{8}C_3 \cdot ^{48}C_{4} = 10,896,480[/tex] different ways.
