Question

the four walls and ceiling of a room are to be painted with five colors available. how many ways can this be done if bordering sides of the room must have different colors?

Answer 1

The total number of ways to paint the room is 120.

Step-by-step explanation:

To find the number of ways to paint the four walls and ceiling of a room with five colors, where bordering sides must have different colors, we can use the concept of permutations. Since there are five available colors, we have five choices for the first wall. For the second wall, we have four colors left to choose from, and for the third wall, we have three colors left. Finally, for the fourth wall, we have two colors left. Therefore, the total number of ways to paint the room is 5 × 4 × 3 × 2 = 120 ways.

Answer 2

To determine the number of ways to paint the four walls and ceiling of a room with five available colors, ensuring adjacent sides have different colors, we apply combinatorics. There are 5 choices for the first wall, then 4 for the second, 3 for the third, 3 for the fourth, and 2 for the ceiling, resulting in 360 combinations.

Step-by-step explanation:

Calculating Combinations for Painting a Room

If the four walls and ceiling of a room are to be painted with five different colors available, and bordering sides of the room must have different colors, we can approach this as a combinatorics problem. The first wall can be painted with any of the five colors. For each choice of color for the first wall, the second wall can be painted with any of the remaining four colors, since it needs to be a different color than the first wall. This reasoning continues for the remaining walls and ceiling, with each subsequent choice having one fewer available color.

Let's assign the order in which the sides will be painted as follows: wall 1, wall 2, wall 3, wall 4, and ceiling. There are 5 choices for wall 1, 4 choices for wall 2, 3 choices for wall 3, 3 choices for wall 4 (since it only borders two walls and the ceiling which has not yet been painted), and finally, 2 choices for the ceiling (since it borders all four walls which have been painted with different colors).

The total number of ways to paint the room can be calculated as the product of these choices:

1. 5 choices for wall 1
2. 4 choices for wall 2
3. 3 choices for wall 3
4. 3 choices for wall 4
5. 2 choices for the ceiling

This results in 5 x 4 x 3 x 3 x 2 = 360 ways to paint the room under the given conditions.