Problem Solving

Solution:
Try to understand this problem through a smaller example.
Suppose there are 5 A's and 5 B's and there are 2 places into which they can be put so that there is a different arrangement each time.

So the possibilities are A A
A B
B A
B B

They are 4 in number.
This can be derived in another way.
The first place can have 2 things, either A or B.
The second place can also have 2 things, either A or B.
So total number of possibilities is 2 * 2 = 4.

Similarly take another problem.
Suppose there are 2 A's, 5 B's and and there are 3 places in which they can be arranged.
So the possibilities are A A B, A B A, B A A, A B B, B A B, B B A, B B B.
These are 7 in number.
The answer can be derived in another way.
First place can have 2 things,A or B.
Second place can have 2 things, A or B.
Third place can also have 2 things, A or B.
So the possible number of arrangements are 2*2*2 = 8.
But we have to omit 1 arrangement where there are 3 A's that is A A A because we have only 2 A's given.
So the final answer is 8 - 1 = 7

For this particular problem there are 6 places and 5 white, 5 black and 5 red blocks.
Or the possible number of arrangements is 3*3*3*3*3*3 - 3 = 3^6 - 3 = 726.

The 3 arrangements that are subtracted are the ones where there are 6 white blocks, 6 black blocks and 6 red blocks because they are all 5 in number.

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)

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
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Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch our free video: https://www.gmatprepnow.com/module/gmat- ... /video/775

Then you can try solving the following questions:

EASY
- https://www.beatthegmat.com/what-should- ... 67256.html
- https://www.beatthegmat.com/counting-pro ... 44302.html
- https://www.beatthegmat.com/picking-a-5- ... 73110.html
- https://www.beatthegmat.com/permutation- ... 57412.html
- https://www.beatthegmat.com/simple-one-t270061.html
- https://www.beatthegmat.com/mouse-pellets-t274303.html


MEDIUM
- https://www.beatthegmat.com/combinatoric ... 73194.html
- https://www.beatthegmat.com/arabian-hors ... 50703.html
- https://www.beatthegmat.com/sub-sets-pro ... 73337.html
- https://www.beatthegmat.com/combinatoric ... 73180.html
- https://www.beatthegmat.com/digits-numbers-t270127.html
- https://www.beatthegmat.com/doubt-on-sep ... 71047.html
- https://www.beatthegmat.com/combinatoric ... 67079.html


DIFFICULT
- https://www.beatthegmat.com/wonderful-p- ... 71001.html
- https://www.beatthegmat.com/ps-counting-t273659.html
- https://www.beatthegmat.com/permutation- ... 73915.html
- https://www.beatthegmat.com/please-solve ... 71499.html
- https://www.beatthegmat.com/no-two-ladie ... 75661.html
- https://www.beatthegmat.com/laniera-s-co ... 15764.html

Cheers,
Brent