# The teacher wants to give some chocolates to the boys of his class. In how many ways can he distribute 10 identical chocolates to 4 boys?

The teacher wants to give some chocolates to the boys of his class. In how many ways can he distribute 10 identical chocolates to 4 boys?

Can someone give an explanation for this answer.

No. of ways = $C_{4-1}^{10+4-1} = 286$

-

## 1 Answer

Convert the question to a "stars & bars" one: put the ten chocolates in a line, represented by stars: $$\star \;\;\star \;\;\star \;\;\star \;\;\star \;\;\star \;\;\star \;\;\star \;\;\star \;\;\star \;\;$$ You now have 3 bars to place, which will mark the boundaries of each boy's share. For example, if the teacher gave the first boy 5 chocolates, the second boy 2, the third boy 1, and the fourth boy 2, you would place the bars like this: $$\star \;\;\star \;\;\star \;\;\star \;\;\star \;|\;\star \;\;\star \;|\;\star \;|\;\star \;\;\star \;\;$$ If the fourth boy got everything, the bars would be like this: $$|\;\;|\;\;|\;\star \;\;\star \;\;\star \;\;\star \;\;\star \;\;\star \;\;\star \;\;\star \;\;\star \;\;\star \;\;$$ Overall, you are looking at the number of ways to have 13 symbols in a row, 10 of which are stars and 3 of which are bars. That explains why the answer is $$\binom{13}{3}=286.$$

-