Bane had 3 different color paints with him - Red, Green, and Blue.

[BTGModeratorVI]

Bane had 3 different color paints with him - Red, Green, and Blue. He wanted to paint a wall with 6 vertical stripes, but no two adjacent stripes could be of the same color. Assuming that Bane can use one color more than once, in how many ways can Bane paint the wall?

A. 32
B. 64
C. 96
D. 243
E. 729
Answer: C
Source: e-GMAT

[deloitte247]

- The wall has 6 vertical shapes
- Two adjacent stripes cannot be of the same order
- Ban can use one color more than once
- The first stripe can be painted with either red, green or blue paint => i.e 3 possible ways to start the first stripe
If stripe 1 = red, stripe 2 will be either green or blue => 2 possible ways
Then, the remaining 4 stripes will be a variation of 2 possible different colors because adjacent stripes must have different colors, and one color can be used twice.
no. of ways = 3 * 2 * 2 * 2 * 2 *2
$$=3\cdot2^5$$
$$=3\cdot32$$
$$=96$$

Answer = option C

[Brent@GMATPrepNow]

Bane had 3 different color paints with him - Red, Green, and Blue. He wanted to paint a wall with 6 vertical stripes, but no two adjacent stripes could be of the same color. Assuming that Bane can use one color more than once, in how many ways can Bane paint the wall?

A. 32
B. 64
C. 96
D. 243
E. 729
Answer: C
Source: e-GMAT

Take the task of painting the 6 stripes and break it into stages.

Stage 1: Select a color for the first stripe
Since we have 3 colors to choose from, we can complete stage 1 in 3 ways

Stage 2: Select a color for the 2nd stripe
This stripe cannot be the same color as stripe #1.
So, there are 2 remaining colors from which to choose, which means we can complete this stage in 2 ways.

Stage 3: Select a color for the 3rd stripe
This stripe cannot be the same color as stripe #2.
So, there are 2 remaining colors from which to choose, which means we can complete this stage in 2 ways.

Stage 4: Select a color for the 4th stripe
Applying the logic we applied above, we can complete this stage in 2 ways

Stage 5: Select a color for the 5th stripe
We can complete this stage in 2 ways

Stage 6: Select a color for the 6th stripe
We can complete this stage in 2 ways.

By the Fundamental Counting Principle (FCP), we can complete all 6 stages (and thus paint all 6 stripes) in (3)(2)(2)(2)(2)(2) ways (= 96 ways)

Answer: C

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch this free 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