Question

In how many ways can a teacher arrange 4 students in the front row of a classroom with a total of 30 students?

Answer 1

arranging depends on the order of the students, so it is a permutation. it's a permutation without repetition since teacher can only pick one student once
\[P(30, 4) = \frac{ 30! }{ (30-4)! }\]

Answer 2

basically it works like this:
1) he can have 30 different students on spot one
2) now he can still have 29 different students on spot two
etc

we multiply the number from 1): "30 different students" with all later possible ways, example: pick a number 1-30, then black or white.

black or white are two choices. however, the total choices are: 30x2, because he can have 1-black, 1-white, 2-black, 2-white, 3-black, 3-white

that's why we will multiply
30 (first choice making a difference) X remaining choices .........

we stop after four choices, since then, there are no more different choices possible. to eliminate all excess from 30! which is
30x29x28x27x26x25x24x23...x4x3x2x1

we divide with all choices we didn't use:
26x25x24x23...x4x3x2x1

and are left with the actual permutation of choices that are done

Answer 3

the answer is 657,720
by permutation