(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.)
(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.
(a) In this case, Magnus can either give a sticker to a friend or not give a sticker to a friend. So each friend has 2 options, and since there are 12 friends, the total number of ways to give out the stickers is $2^{12}=\boxed{4096}$.
(b) In this case, we can think of it as first giving each friend one sticker, and then distributing the remaining 12-12=0 stickers. Since every friend already has a sticker, there are no restrictions on how to distribute the remaining stickers. So the total number of ways is the same as the number of ways to distribute 0 stickers among 12 friends, which is $\binom{12}{0}=\boxed{1}$.
(a) To find the number of ways Magnus can give out 12 identical stickers to 12 friends, we can use the concept of stars and bars.
In this case, 12 stickers can be represented by 12 stars: ************. To divide these stickers among 12 friends, we can use 11 bars to create 12 compartments. For example, placing a bar between the first and second star will represent the first friend receiving 1 sticker, and the second friend receiving 0 stickers.
The number of ways to arrange the 12 stars and 11 bars can be calculated using the formula: (n + k - 1) choose (k - 1), where n is the number of stars and k is the number of bars. In this case, n = 12 and k = 11.
Using the formula, the number of ways Magnus can give out the stickers is:
(12 + 11 - 1) choose (11 - 1) = 22 choose 10 = 22C10 = 45,045 ways.
Therefore, there are 45,045 ways for Magnus to give out 12 identical stickers to 12 of his friends.
(b) In this case, every friend must receive at least one sticker. We can start by giving each friend one sticker, which leaves us with 12 - 12 = 0 stickers remaining.
Now, we need to distribute the remaining 0 stickers among the friends. Since each friend must receive at least one sticker, there are no remaining stickers to distribute. Therefore, there is only one way for Magnus to give out 12 identical stickers to 12 friends, where every friend receives at least one sticker.
To find the number of ways Magnus can give out 12 identical stickers to 12 of his friends, we can use the concept of distributing identical objects to distinct containers (friends).
(a) In this case, there are no restrictions on whether everyone gets a sticker or not. We can think of this problem as distributing 12 identical stickers into 12 distinct containers. Since the stickers are identical, all we need to find is how many ways we can distribute them.
To solve this problem, we can use a concept called "stars and bars." Imagine representing the stickers with 12 stars (************) and representing the friends with 11 bars (| | | | | | | | | | |). We need to arrange the stars and bars in a line, and the number of ways to do so represents the number of ways to distribute the stickers.
There are a total of 11 spaces between the bars where the stars can be placed. We need to choose 12 of these spaces to place the stars, and the remaining spaces will be filled with bars. Using combinations, we can calculate this as:
C(11, 12) = 11! / (12! * (11 - 12)!) = 11! / (12! * (-1)!) = 1/12.
Therefore, there is only one way to distribute the stickers to the friends if not everyone needs to get a sticker.
(b) In this case, every friend must receive at least one sticker. This means we need to make sure none of the containers (friends) are empty.
One approach to solve this problem is to use the concept of "stars and bars" again. We will need to distribute 12 identical stickers among 12 distinct containers, but this time each container must receive at least one sticker.
To ensure that each friend gets at least one sticker, we can give each friend one sticker initially. Now, we are left with 12 - 12 = 0 stickers to distribute. We can apply the same process as before for the remaining 0 stickers.
Using the concept of "stars and bars" again, we have 0 stars (no more stickers) and 11 bars (representing the spaces between friends). The number of ways to arrange these stars and bars is:
C(11, 0) = 11! / (0! * (11 - 0)!) = 11! / (0! * 11!) = 1.
Therefore, there is only one way to distribute the stickers to the friends if everyone needs to get at least one sticker.