2021 AMC 10 A Fall

Solution 1:

For simplicity purposes, we assume that the balls are indistinguishable and the bins are distinguishable. Let be the number of ways to distribute 20 balls into 5 bins. We have Therefore, the answer is Remark By the stars and bars argument, we get

Solution 2:

For simplicity purposes, the balls are indistinguishable and the bins are distinguishable. Let be equal to where is the total number of combinations and is the number of cases where every bin ends up with 4 balls. Notice that we can take 1 ball from one bin and place it in another bin so that some bin ends up with 3 balls, another with 5 balls, and the other three with 4 balls each. We have Therefore, we get , from which