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

by AAPL » Wed Nov 13, 2019 2:38 am
e-GMAT

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. 723

OA C
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by Brent@GMATPrepNow » Wed Nov 13, 2019 6:10 am
AAPL wrote:e-GMAT

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. 723

OA C
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
--------------------------

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by swerve » Thu Nov 14, 2019 6:46 am
AAPL wrote:e-GMAT

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. 723

OA C
Using the rules mentioned in the question STEM we get,

First strip can be painted with 3 colors.
Second strip can be painted with 2 colors.
Third strip can be painted with 2 colors.
Fourth strip can be painted with 2 colors.
Fifth strip can be painted with 2 colors.
Sixth strip can be painted with 2 colors.

Hence the number of ways \(\Rightarrow 3\cdot 2\cdot 2 \cdot 2 \cdot 2 \cdot 2 \Rightarrow 3 \cdot 2^5 \Rightarrow 3\cdot 32 \Rightarrow 96\) ways.
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