In a class of 28 students, 4 prizes are to be distributed. In how many ways can this happen,

a) if the prizes are identical and a student can receive at most one prize?

b) if the prizes are identical and a student can receive multiple prizes?

c) if the prizes are different and a student can receive at most one prize?

d) if the prizes are different and a student can receive multiple prizes?
a) From 28 students, 4 students need to be selected for 4 prizes. Since the prizes are all equal, the order of the selected students does not matter, so we are dealing with combinations, specifically combinations without repetition, because a student cannot receive more than one prize.

Thus,

$$
C_{28}^{4}=\binom{28}{4}=\frac{28 \cdot 27 \cdot 26 \cdot 25}{1 \cdot 2 \cdot 3 \cdot 4}=7 \cdot 9 \cdot 13 \cdot 25=20475
$$

b) The same as the previous case, but now we are dealing with combinations with repetition, so

$$
C_{28}^{i, 4}=\frac{28 \cdot 29 \cdot 30 \cdot 31}{1 \cdot 2 \cdot 3 \cdot 4}=7 \cdot 29 \cdot 5 \cdot 31=31465
$$

c) Since the prizes are all different, the order of the selected students now matters, so we are dealing with variations, specifically variations without repetition, considering that a student can receive at most one prize.

$$
V_{28}^{4}=28 \cdot 27 \cdot 26 \cdot 25=491400
$$

Finally, in the d) case, we are dealing with variations with repetition, so

$$
V_{28}^{i, 4}=28^{4}=784^{2}=614656
$$

Benedek Tomor (Győr, Révai Miklós g. II. o. t.)