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# A rectangular floor measures 2 by 3 meters. There are 5 whit

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Math Expert
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A rectangular floor measures 2 by 3 meters. There are 5 whit [#permalink]

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19 Nov 2009, 12:50
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95% (hard)

Question Stats:

33% (02:31) correct 67% (02:15) wrong based on 48 sessions

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A rectangular floor measures 2 by 3 meters. There are 5 white, 5 black, and 5 red parquet blocks available. Each block measures 1 by 1 meter. In how many different colors patterns can be floor be parqueted?

(A) 104
(B) 213
(C) 3^5
(D) 705
(E) 726

M6-05
[Reveal] Spoiler: OA

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Re: A rectangular floor measures 2 by 3 meters. There are 5 whit [#permalink]

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19 Nov 2009, 21:10
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I get E: 726.

I assumed that the rectangular floor is looked at uni-directionally. So in essence:

[1 2 3]
[4 5 6]

is represented as

[1 2 3 4 5 6]

So 3 ways to select the color of the first tile, 3 ways to select the color of the second tile, etc. Now assuming that there were 6 tiles of each color, you would have:

3 * 3 * 3 * 3 * 3 * 3 = 3^6 = 729 possibilities.

However, these possibilities allow the inclusion of an all black, an all white, and an all red parquet. You need to take away these three distinct possibilities (since we only have 5 tiles of each color).

729 - 3 = 726.

Therefore, there are 726 different color patterns available.
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Re: A rectangular floor measures 2 by 3 meters. There are 5 whit [#permalink]

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19 Nov 2009, 23:23
i agree E should be the correct answer.

3^6 possible patterns - 3 all same color patterns = 729 -3 = 726 different color patterns
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Re: A rectangular floor measures 2 by 3 meters. There are 5 whit [#permalink]

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20 Nov 2009, 01:05
Nice work AKProdigy. I had 3^6 but couldn't figure out where to lose the 3. +1
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A rectangular floor measures 2 by 3 meters. There are 5 whit [#permalink]

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20 Nov 2009, 08:54
AKProdigy87 wrote:
I get E: 726.

I assumed that the rectangular floor is looked at uni-directionally. So in essence:

[1 2 3]
[4 5 6]

is represented as

[1 2 3 4 5 6]

So 3 ways to select the color of the first tile, 3 ways to select the color of the second tile, etc. Now assuming that there were 6 tiles of each color, you would have:

3 * 3 * 3 * 3 * 3 * 3 = 3^6 = 729 possibilities.

However, these possibilities allow the inclusion of an all black, an all white, and an all red parquet. You need to take away these three distinct possibilities (since we only have 5 tiles of each color).

729 - 3 = 726.

Therefore, there are 726 different color patterns available.

This question can be solved with two approaches: the long one and the short and elegant one, as AKProdigy87 proposed. With the later it's definitely possible to solve this problem in 2 mins.

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

Imagine the case in which we have not 5 blocks of each color but 6, then each slot from 2*3=6 would have 3 color choices to be filled with: white, black, or red. That means that total different ways to fill 6 slots would be 3*3*3*3*3*3=3^6;

Now, what is the difference between this hypothetical case and the one in the question? As we allowed 6 blocks of each color instead of 5, then we would get 3 patterns which are impossible when we have 5 blocks of each color: all white, all red and all black. Thus we should subtract these 3 cases: 3^6-3=726.

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Re: A rectangular floor measures 2 by 3 meters. There are 5 whit [#permalink]

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02 Dec 2009, 23:27
That's a very elegant solution. Was wondering - wouldn't there be a need to subtract out cases where 1 red tile is being replaced by another red tile? The colour pattern will be the same in the cases where we replace 1 coloured tile in a pattern with another tile of the same colour.
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Re: A rectangular floor measures 2 by 3 meters. There are 5 whit [#permalink]

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07 Dec 2009, 05:58
Total number of squares we need to fill = 6

Number of colors we have = 3

Therefore, total number of patterns = 3*3*3*3*3*3 = 729

However, this is considering that we can have a case in which all tiles are the same color. Since we are given that the quantity of each tile is 5 and the number of tiles required is 6, we know that this case cannot be possible. Therefore we must subtract the cases in which all tiles will be of the same color (3 cases since there are 3 colors).

Thus our answer should be : 729 - 3 = 726

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Permutation and Combination : Tiles arrangement [#permalink]

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12 Jul 2014, 00:34
Can some one solve this question and post his/her detailed solution ? I found this question in gmat club question tests. I didn't understand the explanation thorouhly though
Question : 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?
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C'est la vie

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Re: Permutation and Combination : Tiles arrangement [#permalink]

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12 Jul 2014, 02:05
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mshrek wrote:
Can some one solve this question and post his/her detailed solution ? I found this question in gmat club question tests. I didn't understand the explanation thorouhly though
Question : 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?

as shown in the figure, we have a space of 6 tiles available with us. each measuring 1 by 1 meter. lets name these spaces as 1,2,3,4,5 and 6. In first (1) of these six spaces any of the 3 colors can occur . similarly for space 2,3,4,5 and 6 we have 3 options. now since each of three available colors has only 5 tiles, therefore we cannot fill the space with 6 tiles of same color. hence we must subtract three cases in which we have assumed that 6 spaces are filled with one of the three colors.

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Math Expert
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Posts: 34420
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Kudos [?]: 79415 [0], given: 10016

Re: A rectangular floor measures 2 by 3 meters. There are 5 whit [#permalink]

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12 Jul 2014, 03:40
mshrek wrote:
Can some one solve this question and post his/her detailed solution ? I found this question in gmat club question tests. I didn't understand the explanation thorouhly though
Question : 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?

Merging similar topics. Please refer to the discussion above.
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Re: A rectangular floor measures 2 by 3 meters. There are 5 whit   [#permalink] 12 Jul 2014, 03:40
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