Question
Find the number of ways that Magnus can give out $9$ identical stickers to $2$ of his friends, if every friend gets at least one sticker.
Answers
GPT 3.5
We can use stars and bars to solve this problem.
Since each friend must receive at least one sticker, we can give one sticker to each friend initially, leaving $7$ stickers left to distribute. Now, we have $7$ indistinguishable objects (the stickers) and $2$ distinguishable containers (the friends).
By stars and bars, the number of ways to distribute the $7$ stickers among the $2$ friends is ${7+2-1 \choose 2-1} = {8 \choose 1} = 8$.
Therefore, Magnus can give out the $9$ stickers in $8$ different ways.
Since each friend must receive at least one sticker, we can give one sticker to each friend initially, leaving $7$ stickers left to distribute. Now, we have $7$ indistinguishable objects (the stickers) and $2$ distinguishable containers (the friends).
By stars and bars, the number of ways to distribute the $7$ stickers among the $2$ friends is ${7+2-1 \choose 2-1} = {8 \choose 1} = 8$.
Therefore, Magnus can give out the $9$ stickers in $8$ different ways.