Combinations

swerve wrote: ↑

Thu Dec 17, 2020 5:34 am

Jevan must paint 3 rooms in a house. Room A can be painted orange, red or green. Room B can be painted orange, white or red. Room C can be painted white, red or green. The 3 rooms cannot all be painted the same color. In how many different ways could Jevan paint the 3 rooms?

A. 24
B. 26
C. 27
D. 63
E. 64

The OA is B

Source: Magoosh

Take the task of painting the 3 rooms and break it into stages.

Stage 1: Select a color for Room A
Room A can be painted orange, red or green
So, we can complete stage 1 in 3 ways

Stage 2: Select a color for Room B
Room B can be painted orange, white or red.
So we can complete this stage in 3 ways.

Stage 3: Select a color for Room C
Room C can be painted white, red or green.
So we can complete this stage in 3 ways.

By the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus paint all 3 rooms) in (3)(3)(3) ways (= 27 ways)

However, some of these 27 possible outcomes BREAK the condition that the 3 rooms cannot all be painted the same color.
Given the different color options for each room, we can see that the ONLY way that the 3 rooms can be the same color is when all 3 rooms are painted RED.
Since there is only 1 way to BREAK the given condition, the number of GOOD outcomes = 27 - 1 = 26

Answer: B
Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch this video: https://www.gmatprepnow.com/module/gmat- ... /video/775

You can also watch a demonstration of the FCP in action: https://www.gmatprepnow.com/module/gmat ... /video/776

Then you can try solving the following questions:

EASY
- https://www.beatthegmat.com/what-should- ... 67256.html
- https://www.beatthegmat.com/counting-pro ... 44302.html
- https://www.beatthegmat.com/picking-a-5- ... 73110.html
- https://www.beatthegmat.com/permutation- ... 57412.html
- https://www.beatthegmat.com/simple-one-t270061.html


MEDIUM
- https://www.beatthegmat.com/combinatoric ... 73194.html
- https://www.beatthegmat.com/arabian-hors ... 50703.html
- https://www.beatthegmat.com/sub-sets-pro ... 73337.html
- https://www.beatthegmat.com/combinatoric ... 73180.html
- https://www.beatthegmat.com/digits-numbers-t270127.html
- https://www.beatthegmat.com/doubt-on-sep ... 71047.html
- https://www.beatthegmat.com/combinatoric ... 67079.html


DIFFICULT
- https://www.beatthegmat.com/wonderful-p- ... 71001.html
- https://www.beatthegmat.com/permutation- ... 73915.html
- https://www.beatthegmat.com/permutation-t122873.html
- https://www.beatthegmat.com/no-two-ladie ... 75661.html
- https://www.beatthegmat.com/combinations-t123249.html


Cheers,
Brent