Question

each of ten houses on a street (five houses on each side) is painted brown, yellow or white. each house is painted only one color and each color is used on at least one house. no two colors are used to paint the same number of houses. in how many ways could the ten houses on the street be painted?

Answer 1

There are different options for painting the houses on the street, and the total number of ways can be found by calculating permutations for each option and summing them up.

Step-by-step explanation:

There are 10 houses on the street, with 5 on each side. Each house can be painted brown, yellow, or white. In order to find the number of ways the houses can be painted, we can use a combination of combinatorics and permutations.

First, we need to determine how many houses can be painted with each color. Since no two colors can be used to paint the same number of houses, we have the following options:

1. 1 house of one color, 4 houses of another color, and 5 houses of the remaining color.
2. 2 houses of one color, 3 houses of another color, and 5 houses of the remaining color.
3. 3 houses of one color, 2 houses of another color, and 5 houses of the remaining color.

For each option, we need to calculate the number of ways the houses can be painted using permutations. For example, if we have 1 brown house, 4 yellow houses, and 5 white houses, we can arrange them in 10!/(1!4!5!) = 10,080 ways.

Similarly, we calculate the number of ways for the other options and sum them up to get the total number of ways the houses can be painted on the street.