# How many ways can the eggs be distributed among the eight children, with no restriction on the number of eggs each child receives?

A basket contains eight green, seven yellow, and four purple eggs. (Assume eggs of the same colour are indistinguishable.) These eggs are all handed out to a group of eight children.

(a)How many ways can the eggs be distributed among the eight children, with no restriction on the number of eggs each child receives?

(b)Let n be the largest number of eggs that any child receives. What is the range of values that n can be? (Find the largest a and the smallest b that we can determine such that we are guaranteed $$a ≤ n ≤ b$$.)

So I am trying to figure this question out. There are a total of 19 eggs, with 3 colours. The eggs of the same colour are indistinguishable - so this means that the orders won't matter. But how would I construct a combination for this? I have been stuck on this question for a while and I have no progress. Any hints will be appreciated.

For (b), since there are 19 eggs, would the largest be $$11$$ and the smallest be 1, so $$1 ≤ n ≤ 11$$?

They wanted $$a$$ to be the largest value but it is on the left side of $$n$$...

• Hint: Treat each color of eggs separately. That gives you three combinations with repetition problems to solve. – N. F. Taussig Mar 21 at 0:42
• For (b), I interpret $n$ to be the number of eggs received by the child who had the most eggs. Then if one child gets all the eggs $n=19$, where if the eggs are split as evenly as possible the child with most eggs will get $n=3$. They want $a$ as a lower bound, so $a \le n$, but the point of a lower bound is to be as high as possible (e.g. $-10000$ is also a lower bound but is uninteresting), which makes it as tight as possible. Hence they want the maximum $a$ which is still guaranteed to be $a \le n$. – antkam Mar 22 at 19:42