(a) Find the number of ways that Magnus can give out 12 identical stickers to 12 of his friends. (Not everyone has to get a sticker.)

Question

142,506 ways - with no restrictions.

(b) Find the number of ways that Magnus can give out 12 identical stickers to 12 of his friends, if every friend gets at least one sticker.
12,650 ways

Answer 1

To solve part (a), we can think of each sticker as being able to go to one of the 12 friends. Since the stickers are identical, the order in which they are given out does not matter.

This problem can be solved using the stars and bars method. We can think of the 12 friends as "stars" and the 12 stickers as "bars" that divide the friends into groups. For example, if Magnus gives 3 stickers to the first friend, 2 stickers to the second friend, and 1 sticker to the third friend, we can represent this as follows:
* * * | * * | *
The number of ways to arrange the stickers and friends is then equivalent to the number of ways to arrange the 12 stars and 11 bars. This is given by the binomial coefficient C(12+11, 11), which can be calculated as:

C(23, 11) = (23!/11!12!) = 23! / (11!12!) = (23 * 22 * 21 * ... * 12) / (11 * 10 * 9 * ... * 1) = 142,506

Therefore, there are 142,506 ways for Magnus to give out the stickers without any restrictions.

For part (b), since every friend must receive at least one sticker, we can treat this as the previous problem but subtract the cases where one or more friends do not receive a sticker.

Let's consider the cases where at least one friend does not receive a sticker. There are 12 possible friends who could be left without a sticker. We can subtract the number of ways to distribute the stickers in these cases from the total number of ways without any restrictions.

If one friend does not receive a sticker, there are 12 ways to choose which friend that is. Then, we can think of the remaining 11 stickers as being able to go to the remaining 11 friends, which can be arranged in (11+10, 10) = 21! / (11!10!) = 21 ways.

If two friends do not receive a sticker, there are C(12, 2) = 66 ways to choose which friends those are. Then, we can think of the remaining 10 stickers as being able to go to the remaining 10 friends, which can be arranged in (10+9, 9) = 19! / (10!9!) = 19 ways.

We can continue this pattern for cases where 3 or more friends do not receive a sticker, but notice that as the number of friends without a sticker increases, the number of ways for the remaining stickers to be distributed decreases. In fact, there will be no cases where more than 3 friends are left without a sticker, as giving each friend at least one sticker would require at least 10 stickers, leaving at most 2 stickers to be shared among 2 friends.

Therefore, the total number of ways for Magnus to give out the stickers, with each friend receiving at least one sticker, is:

142,506 - (12 * 21) - (66 * 19) = 12,650

Therefore, there are 12,650 ways for Magnus to give out the stickers with each friend receiving at least one sticker.