I had this problem in a local math competition and was wondering how to solve it, even though I know the answer.
Bob has 24 apples. He want to give some apples to two of his friends Jill and Patty. He wants each of the three friends (counting himself) to end up with at least 2 apples each. How many possible ways could he distribute these apples (without breaking any)?
I tried permutations, combinations, and the multiplication rule but to no avail. Help would be greatly appreciated.
Thank you in advance!
If each person gets $2$ apples, then there remain $18$ apples to be distributed. Let $X_1, X_2, X_3$ be the number of additional apples that the first, second, and third friends get, respectively.
With these definitions, we wish to find the number of nonnegative integer valued vector solutions $(X_1,X_2,X_3)$ to $X_1 + X_2 + X_3 = 18$, $X_1, X_2, X_3 \geq 0$.
The result is well known and often referred to as the (nonnegative) solution to the "stars and bars" problem: ${18 + 3 - 1} \choose {3-1}$ $=$ $20 \choose 2$ $=190$ ways.