+1839 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?
+1768 We can solve this problem by considering the different cases for how many candies the twins can get:
Case 1: Twins get 0 candies each
In this case, we essentially have 3 children (excluding the twins) who can receive 3 identical candies.
Distributing 3 candies to 3 children can be done in (3 + 3 - 1)! / (3 - 1)! = 6 ways using the stars and bars method (where 3 stars represent candies and 2 bars represent dividers separating the children).
Case 2: Twins get 1 candy each
Now, we have to distribute 1 candy to each twin (which is fixed) and 1 remaining candy to the 3 remaining children.
Distributing 1 candy to 3 children can be done in 3 ways.
Case 3: Twins get 2 candies each
Here, the twins get a total of 4 candies (2 each), leaving 1 candy for the remaining child.
There's only 1 way to distribute this remaining candy.
Total Ways
To find the total number of ways to distribute the candies, we add the number of ways for each case:
Total Ways = Cases (Twins get 0) + Cases (Twins get 1) + Cases (Twins get 2)
Total Ways = 6 + 3 + 1 = 10
Therefore, there are 10 ways to distribute the 3 candies such that the twins get an equal amount.
