Permutation

ronnie1985 wrote:A rectangular floor measures 2 by 3 meters. There are 5 white, 5 black, and 5 red parquet blocks available. If each block measures 1 by 1 meter, in how many different color patterns can the floor be parqueted?
A. 104
B. 213
C. 577
D. 705
E. 726

OA [spoiler](E)[/spoiler]

Seeking explanatin

There are six spots in the 2x3 floor. We'll label them as #1, 2, 3, 4, 5, and 6.
Each of the six spots can be white, black, or red. So, we'll take the task of placing a block in each spot and break it into six stages.

Stage 1: Select a colored block for space #1
Stage 2: Select a colored block for space #2
Stage 3: Select a colored block for space #3
Stage 4: Select a colored block for space #4
Stage 5: Select a colored block for space #5
Stage 6: Select a colored block for space #6

Now we'll determine the number of ways to complete each stage.

Stage 1: There are are 3 colors to choose from, so we can accomplish this stage in 3 ways.
Stage 2: There are are 3 colors to choose from, so we can accomplish this stage in 3 ways.
Stage 3: There are are 3 colors to choose from, so we can accomplish this stage in 3 ways.
.
.
Stage 6: There are are 3 colors to choose from, so we can accomplish this stage in 3 ways.

So, the total number of ways to complete all 6 stages = 3x3x3x3x3x3 = 729

Aside: 729 would have been a great distractor for this question.

Important: Notice that my method allows for the possibility of all 6 blocks being the same color. However, since there are only 5 blocks of each color, we can't have all 6 blocks the same color.

So, we need to subtract from 729 all of the arrangements where all 6 blocks are the same color.
Well, there are 3 such arrangements: 1) all blocks white, 2) all blocks black, and 3) all blocks red.

When we subtract the 3 impossible arrangements from 729, we get[spoiler] 726 (E)[/spoiler]

Cheers,
Brent

PS: This solution applies something called the Fundamental Counting Principle (FCP). For more information on the FCP, see: https://www.gmatprepnow.com/module/gmat-counting?id=775

ronnie1985 wrote:A rectangular floor measures 2 by 3 meters. There are 5 white, 5 black, and 5 red parquet blocks available. If each block measures 1 by 1 meter, in how many different color patterns can the floor be parqueted?
A. 104
B. 213
C. 577
D. 705
E. 726

OA [spoiler](E)[/spoiler]

Seeking explanatin

The rectangular floor measures 2 by 3 meters.
Thus there are 2*3 = 6 blocks of measurement 1 by 1 meter.

Now there are 3 possible color for each block.
Thus if we had infinite numbers of parquet blocks of each color, we would've done the decoration in 3^6 = 729 ways.

But we have a limited number of parquet blocks of each color, i.e 5 of each. Therefore all of the blocks cannot be of the same color at the same time. Thus all of the 6 blocks are white or black or red is not possible. Therefore except these 3 impossible cases the scenario is same as if we have infinite numbers of parquet blocks.

Therefore, actual number of different patterns = (729 - 3) = 726

The correct answer is E.