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# Rectangular floor patterns

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Math Expert
Joined: 02 Sep 2009
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Rectangular floor patterns [#permalink]  19 Nov 2009, 11:50
Expert's post
00:00

Difficulty:

5% (low)

Question Stats:

40% (01:01) correct 60% (03:42) wrong based on 5 sessions
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
[Reveal] Spoiler: OA

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Manager
Joined: 11 Sep 2009
Posts: 129
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Kudos [?]: 142 [6] , given: 6

Re: Rectangular floor patterns [#permalink]  19 Nov 2009, 20:10
6
KUDOS
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.
Intern
Joined: 14 Aug 2009
Posts: 8
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Kudos [?]: 2 [0], given: 1

Re: Rectangular floor patterns [#permalink]  19 Nov 2009, 22:23
i agree E should be the correct answer.

3^6 possible patterns - 3 all same color patterns = 729 -3 = 726 different color patterns
Senior Manager
Joined: 31 Aug 2009
Posts: 423
Location: Sydney, Australia
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Kudos [?]: 91 [0], given: 20

Re: Rectangular floor patterns [#permalink]  20 Nov 2009, 00:05
Nice work AKProdigy. I had 3^6 but couldn't figure out where to lose the 3. +1
Math Expert
Joined: 02 Sep 2009
Posts: 17317
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Kudos [?]: 18380 [0], given: 2348

Re: Rectangular floor patterns [#permalink]  20 Nov 2009, 07:54
Expert's post
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.

+1.

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Joined: 04 Nov 2009
Posts: 64
WE 1: Research
WE 2: Corporate Strat
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Kudos [?]: 13 [0], given: 1

Re: Rectangular floor patterns [#permalink]  02 Dec 2009, 22:27
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.
Senior Manager
Joined: 05 Oct 2008
Posts: 275
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Kudos [?]: 24 [0], given: 22

m 16 - combinations [#permalink]  07 Dec 2009, 04:10
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?

* 104
* 213
* 577
* 705
* 726
Manager
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Kudos [?]: 520 [0], given: 18

Re: m 16 - combinations [#permalink]  07 Dec 2009, 04:58
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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Re: m 16 - combinations   [#permalink] 07 Dec 2009, 04:58
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