homeschool girl
Junior Member
- Joined
- Feb 6, 2020
- Messages
- 123
The Problem:
"Peter purchased a pentagonal pen for his puppy Piper. Now Peter wants to decorate the new pen for Piper, and he would like to paint each side of the pen either red, green, or blue so that each wall is a solid color.
Peter can only paint at night when Piper is sleeping, and unfortunately, it is too dark for him to determine which color he is painting. So for each wall, Peter randomly chooses a can of paint and paints the wall in that color. In the morning, Peter observes the resulting color scheme. The vertices of the pentagon are labeled with the letters [MATH]A, B, C, D,[/MATH] and [MATH]E[/MATH], and these labels are clearly visible during the daytime. What is the probability that no two adjacent walls of the pen have the same color?"
My Answer:
(we "lock" the first wall on a colour then multiply the end product by [MATH]3[/MATH])
there is [MATH]1[/MATH] ways to paint the first wall, [MATH]2[/MATH] ways to paint the second wall because you can't it paint the same colour as the first wall, [MATH]2[/MATH] for the third wall, [MATH]1[/MATH] for the fourth, and [MATH]1[/MATH] option for the fifth wall
giving [MATH]2\cdot2\cdot3=12[/MATH] ways to paint the walls with restrictions.
and there are [MATH]3^5=243[/MATH] ways to paint the walls with no restrictions, so the answer is [MATH]\frac{12}{243}=\boxed{\frac{4}{81}}.[/MATH]
is this right?
"Peter purchased a pentagonal pen for his puppy Piper. Now Peter wants to decorate the new pen for Piper, and he would like to paint each side of the pen either red, green, or blue so that each wall is a solid color.
Peter can only paint at night when Piper is sleeping, and unfortunately, it is too dark for him to determine which color he is painting. So for each wall, Peter randomly chooses a can of paint and paints the wall in that color. In the morning, Peter observes the resulting color scheme. The vertices of the pentagon are labeled with the letters [MATH]A, B, C, D,[/MATH] and [MATH]E[/MATH], and these labels are clearly visible during the daytime. What is the probability that no two adjacent walls of the pen have the same color?"
My Answer:
(we "lock" the first wall on a colour then multiply the end product by [MATH]3[/MATH])
there is [MATH]1[/MATH] ways to paint the first wall, [MATH]2[/MATH] ways to paint the second wall because you can't it paint the same colour as the first wall, [MATH]2[/MATH] for the third wall, [MATH]1[/MATH] for the fourth, and [MATH]1[/MATH] option for the fifth wall
giving [MATH]2\cdot2\cdot3=12[/MATH] ways to paint the walls with restrictions.
and there are [MATH]3^5=243[/MATH] ways to paint the walls with no restrictions, so the answer is [MATH]\frac{12}{243}=\boxed{\frac{4}{81}}.[/MATH]
is this right?
