+311 1. There is a group of five children, where two of the children are twins. In how many ways can I distribute 8 identical pieces of candy to the children, if the twins must get an equal amount of candy?
2. In how many ways can you distribute 8 indistinguishable balls among 5 distinguishable boxes, if at least one of the boxes must be empty?
+729 Problem 1:
Let's consider the twins as one entity for the distribution of candy. This reduces the problem to distributing 8 identical pieces of candy to 4 entities (3 individual children and one entity representing the twins).
We can use the stars and bars method to solve this problem. There are 8 candies and 3 "dividers" to separate them into 4 entities. The number of ways to distribute the candies is then:
C(8 + 3, 3) = C(11, 3) = 165
Therefore, there are 165 ways to distribute the 8 pieces of candy to the children, with the twins receiving an equal amount.
+729 Problem 2:
There are many ways to approach this problem, but one way to think about it is to use a combination of distributions with exactly one box empty and distributions with more than one box empty.
If one box must be empty, there are 5 ways to choose which box will be empty. Then, we need to distribute the remaining 8 balls among the 4 remaining boxes. This can be done using stars and bars, which is a method used to count the number of ways to distribute indistinguishable objects into distinguishable boxes.
Using stars and bars, the number of ways to distribute 8 balls among 4 boxes is (8+4-1 choose 4-1) = 11 choose 3 = 165.
Therefore, the total number of ways to distribute 8 balls among 5 boxes, with at least one box empty, is 5 * 165 = 825 ways.
