Question

The Boxer and the Baller

How many ways are there to distribute seven balls into five boxes, where each box must have
at least one ball in it if
a both the balls and boxes are unlabeled?

Answer 1

There are 2 distinct ways to distribute seven identical balls into five identical boxes, each containing at least one ball, considering that both balls and boxes are unlabeled.

Step-by-step explanation:

The question involves a classic combinatorics problem where we are distributing unlabeled balls into unlabeled boxes such that each box must contain at least one ball. This is a type of integer partitioning problem. We need to find the number of ways to partition seven identical balls into five identical parts where each part has at least one ball. The five parts represent the five boxes, and the balls represent the items to be distributed.

Because each box must contain at least one ball, we start by placing one ball in each of the five boxes. This utilizes five of our seven balls, leaving us with two balls to distribute. These two balls can be distributed amongst the five boxes in the following ways:

• Both balls in one box: There is 1 way to do this since the boxes are identical.
• One ball in each of two different boxes: This case also corresponds to 1 way because the boxes are identical.

Therefore, there are 2 distinct ways to distribute seven identical balls into five identical boxes, where each box has at least one ball.