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by vipulgoyal » Thu Oct 24, 2013 3:13 am
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
perhaps I got the solution but went crazy while doing it using combinatrics
Solution:
3^6, or 729
you have to rule out the combinations of all 6 being white, black or red, so 729 - 3 = 726.
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5c5*5c1*3c2 + 5c4*5c1*5c1*3c2 + 5c3*5c2*5c1*3! + 5c2*5c2*5c2 , even the last one fragment comes out to be 1000 which is far more then right ans
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by lunarpower » Thu Oct 24, 2013 3:31 am
For pretty much all combinatorics problems:

* If there are only a small number of possibilities, just make lists, and count stuff.
(e.g., OG quant supplement #132)

* If there are too many things to list and count.. is it relatively straightforward to calculate the number of possibilities directly?
- If so, do it.
- If not, think about the probability of the opposite event.

In this case, it would be absolutely horrible to calculate the probability directly. The opposite event, though, is "everything in the world except all red, all white, or all black". That's not so bad.
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by GMATGuruNY » Thu Oct 24, 2013 3:35 am
vipulgoyal 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
Floor area = b*h = 2*3 = 6 sq. meters.
Area of each block = b*h = 1*1 = 1 sq. meters.
Total number of blocks required = Floor area/Block area = 6/1 = 6 blocks.

Good combinations = Total possible combinations - Bad combinations

Total possible combinations:
For each block, there 3 options: white, black, or red.
Since there are 3 options for each of the 6 blocks, we get:
3*3*3*3*3*3 = 729.

Bad combinations:
Since there are ONLY 5 of each color, all 6 blocks CANNOT be of the same color.
Thus, there are 3 bad combinations:
All white, all black, all red.

Good combinations = Total - Bad = 729-3 = 726.

The correct answer is E.
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by Brent@GMATPrepNow » Thu Oct 24, 2013 6:41 am
vipulgoyal 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
There are six spots on the 2x3 floor. We'll label them as #1, 2, 3, 4, 5, and 6.
Take the task of placing a block in each spot and break it into stages.

Stage 1: Select a colored block for space #1
There are 3 colors to choose from, so we can complete stage 1 in 3 ways

Stage 2: Select a colored block for space #2
There are 3 colors to choose from, so we can complete stage 2 in 3 ways

Stage 3: Select a colored block for space #3
There are 3 colors to choose from, so we can complete stage 3 in 3 ways
.
.
.
Stage 6: Select a colored block for space #6
There are 3 colors to choose from, so we can complete stage 6 in 3 ways

By the Fundamental Counting Principle (FCP), we can complete all 6 stages (and thus place 6 blocks) in (3)(3)(3)(3)(3)(3) ways (= 729 ways)

Aside: For more information about the FCP, watch our free video: https://www.gmatprepnow.com/module/gmat-counting?id=775

IMPORTANT: This method allows for the possibility that 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 the 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 726

Answer: E

Cheers,
Brent
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