(a) Find the number of ways that Magnus can give out 20 identical stickers to 5 of his friends. (Not everyone has to get a sticker.)

(b) Find the number of ways that Magnus can give out 20 identical stickers to 5 of his friends, if every friend gets at least one sticker.

Guest Dec 30, 2022

#1**+1 **

This is a pretty classic problem.

First, we can set variables for how many stickers each of his friends get. Suppose the first friend is x1, his second friend is x_{2}, and so on, all the way to the fifth friend.

Then, x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=20

Problem A) We can now use stars and bars, the number of non-negative solutions to this equation is (20+5-1) choose (5-1) which is 24 choose 4 which is **10626**

Problem B) We can use stars and bars again, the number of positive integer solutions to the equation, x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=20 is (20-1) choose (5-1) which is **3876 :)**

If you want to learn more about stars and boxes, google is a good place.

hairyberry Dec 30, 2022