Question

(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.

Answer 1

(a) In this case, Magnus can choose any number of friends to give stickers to, from 0 to all 12 friends. This is equivalent to finding the number of subsets of a set of 12 elements, which is $2^{12} = \boxed{4096}$.

(b) Now, every friend must get at least one sticker. This is equivalent to finding the number of ways to distribute the remaining $12-1=11$ stickers among the 12 friends, which is a stars and bars problem with 11 stars and 12-1=11 bars. The number of ways to arrange the stars and bars is $\binom{11+12-1}{12-1} = \binom{22}{11} = \boxed{705432}$.