Question

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

To distribute seven balls into five boxes, where each box must have at least one ball, we can use the stars and bars method. The number of ways to do this can be found using combinations.

Step-by-step explanation:

To distribute seven balls into five boxes, where each box must have at least one ball, we can use the stars and bars method. We can think of the balls as stars and the boxes as bars.

We need to place four bars between the stars to divide them into five boxes. The number of ways to do this can be found using combinations.

The formula for distributing indistinguishable objects into distinguishable boxes with at least one object in each box is (n-1)C(k-1), where n is the number of objects and k is the number of boxes. In this case, n = 7 and k = 5.

Using the formula, we have (7-1)C(5-1) = 6C4 = 6 ways to distribute the balls into the boxes.