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Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by Kim9876Zey » Sun Jun 24, 2012 11:38 am
To furnish a room in a model home, an interior decorator is to select 2 chairs and 2 tables from a collection of chairs and tables in a warehouse that are all different from eachother. If there are 5 chairs in the warehouse and if 150 different combinations are possible, how many tables are in the warehouse?

6 correct
8
10
15
30
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by tutorphd » Sun Jun 24, 2012 1:34 pm
Selecting 2 chairs out of 5: use two slots 5 x 4 = 20 possible permutations but divide by 1 x 2 = 2! because the order doesn't matter, which gives 10 possbile combinations.

Selecting 2 tables out of n (unknown): use two slots giving n(n-1) possible permutations but divide again by 2! because the order of selection doesnt matter, which gives n(n-1)/2 possible combinations.

Combine the two independent selections of chairs and tables by multiplying their possibilities:

10n(n-1)/2 = 150

Simplify: n(n-1) = 30
This is a quadratic equation in n. You can open the brackets and try to factorize it to find the roots but way faster is to simply plug in the answer choices and see which one work. The answer 6 gives 6 x 5 = 30 and is the correct one.

Alternatively, if you don't like algebra, you could have started plugging the anwers from the very begining. You would go through the same reasoning for the talbes and chairs but with numbers not variables like n and will see that only the answer 6 gives the target of 150 total combinations.
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by Brent@GMATPrepNow » Mon Jun 25, 2012 7:20 am
Kim9876Zey wrote:To furnish a room in a model home, an interior decorator is to select 2 chairs and 2 tables from a collection of chairs and tables in a warehouse that are all different from eachother. If there are 5 chairs in the warehouse and if 150 different combinations are possible, how many tables are in the warehouse?

6 correct
8
10
15
30
Total # of combinations = (# of ways to select 2 chairs)(# of ways to select 2 tables)

# of ways to select 2 chairs
5 tables, choose 2 of them.
This can be accomplished in 5C2 ways (10 ways)

Total # of combinations = (# of ways to select 2 chairs)(# of ways to select 2 tables)
150 = (10)(# of ways to select 2 tables)
(# of ways to select 2 tables) = 15

# of ways to select 2 tables
Let N = # of tables.
We have N tables, choose 2.
This can be accomplished in NC2 ways
So, NC2 = 15

From here, we can just start checking answer choices.
We get 6C2 = 15, so there are 15 tables.

If anyone is interested, I have a free video on calculating combinations in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789

Cheers,
Brent
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by gmattesttaker2 » Sat Oct 06, 2012 1:19 pm
Brent@GMATPrepNow wrote:
Kim9876Zey wrote:To furnish a room in a model home, an interior decorator is to select 2 chairs and 2 tables from a collection of chairs and tables in a warehouse that are all different from eachother. If there are 5 chairs in the warehouse and if 150 different combinations are possible, how many tables are in the warehouse?

6 correct
8
10
15
30
Total # of combinations = (# of ways to select 2 chairs)(# of ways to select 2 tables)

# of ways to select 2 chairs
5 tables, choose 2 of them.
This can be accomplished in 5C2 ways (10 ways)

Total # of combinations = (# of ways to select 2 chairs)(# of ways to select 2 tables)
150 = (10)(# of ways to select 2 tables)
(# of ways to select 2 tables) = 15

# of ways to select 2 tables
Let N = # of tables.
We have N tables, choose 2.
This can be accomplished in NC2 ways
So, NC2 = 15

From here, we can just start checking answer choices.
We get 6C2 = 15, so there are 15 tables.

If anyone is interested, I have a free video on calculating combinations in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789

Cheers,
Brent

Hello Brent,

Hope all is well.

Assuming we have 3 chairs that are different say C1, C2 and C3. And we have to pick 2 chairs can we pick them as follows?

C1 C2
C1 C3
C2 C3
C2 C1
C3 C1
C3 C2

Or is C2 C1 the same as C1 C2. I was just getting a bit confused here since with C2 C1 we first pick C2 and then C1 while with C1 C2 we pick the other way round.

Thanks a lot for your help.

Best Regards,
Sri
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by Brent@GMATPrepNow » Sat Oct 06, 2012 4:50 pm
gmattesttaker2 wrote: Hello Brent,

Hope all is well.

Assuming we have 3 chairs that are different say C1, C2 and C3. And we have to pick 2 chairs can we pick them as follows?

C1 C2
C1 C3
C2 C3
C2 C1
C3 C1
C3 C2

Or is C2 C1 the same as C1 C2. I was just getting a bit confused here since with C2 C1 we first pick C2 and then C1 while with C1 C2 we pick the other way round.

Thanks a lot for your help.

Best Regards,
Sri
Hey Sri,

Things are going well, thanks :-)
I hope you're the same.


The question suggests that the order doesn't matter here.
That is, selecting Chair2 then Chair1 is the same as selecting Chair1 then Chair2.

Cheers,
Brent
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