Counting

(a) This is a classic stars and bars problem, where we have 20 identical objects (stickers) that we want to distribute among 5 people. We can model this by placing 4 dividers (bars) among the stickers to separate the stickers into 5 groups. For example, if we have 20 stickers and we place the dividers like this:

****|***|***|****|***

then the first friend gets 4 stickers, the second friend gets 3 stickers, the third friend gets 3 stickers, the fourth friend gets 4 stickers, and the fifth friend gets 6 stickers.

Thus, we need to count the number of ways to place 4 dividers among 20 stickers, which can be done in (20 + 4) choose 4 = 24,024 ways.

(b) This is similar to part (a), but now we need to ensure that every friend gets at least one sticker. We can do this by giving each friend one sticker to start with, and then distributing the remaining 15 stickers using stars and bars. Specifically, we can place 4 dividers among the 15 stickers to separate them into 5 groups, where each group corresponds to the additional stickers given to each friend. For example, if we have 20 stickers and we give each friend one sticker, we are left with 15 stickers, which we can distribute as follows:

*|**|***|****|*****

This corresponds to the first friend getting 2 stickers, the second friend getting 3 stickers, the third friend getting 4 stickers, the fourth friend getting 5 stickers, and the fifth friend getting 6 stickers.

Thus, we need to count the number of ways to place 4 dividers among 15 stickers, which can be done in (15 + 4) choose 4 = 3,432 ways.

Therefore, the number of ways to distribute 20 identical stickers to 5 people, where not everyone has to get a sticker, is 24,024, while the number of ways to distribute 20 identical stickers to 5 people, where every person gets at least one sticker, is 3,432.