There is a group of five children, where two of the children are twins. In how many ways can I distribute $3$ identical pieces of candy to the children, if the twins must get an equal amount of candy?

Answer 1

Let's break this problem down step by step.

First, we need to decide how many pieces of candy each twin will get. Since the twins must get an equal amount of candy, we have two options: each twin gets either $0$ or $1$ piece of candy.

Case 1: Each twin gets $0$ pieces of candy
In this case, the remaining $3$ pieces of candy must be distributed among the other three children (non-twins). This can be done in $C(3+3-1,3) = C(5,3) = 10$ ways.

Case 2: Each twin gets $1$ piece of candy
In this case, each twin already has $1$ piece of candy, and the remaining $1$ piece of candy must be distributed among the other three children. This can be done in $C(1+3-1,3) = C(3,3) = 1$ way.

So, the total number of ways to distribute $3$ pieces of candy to the children, given that the twins must get an equal amount of candy, is $10 + 1 = 11$ ways.