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# At an art gallery, there are 7 spaces to hang 14 paintings.

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At an art gallery, there are 7 spaces to hang 14 paintings. [#permalink]  09 Apr 2019, 09:03
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At an art gallery, there are 7 spaces to hang 14 paintings. The 14 paintings to choose from are: three by Cezanne, three by Monet, one by Renoir, two by Van Gogh, two by Pissarro, two by Degas, and one by Manet.

The even-numbered spaces must hold either a Cezanne or a Monet. The first space must hold a Renoir. The fifth space must hold a Manet, and the remaining spaces must hold either a Van Gogh, a Pissarro, or a Degas.

How many unique arrangements of paintings are possible?

A. 28

B. 72

C. 1200

D. 3600

E. 7776
[Reveal] Spoiler: OA

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Re: At an art gallery, there are 7 spaces to hang 14 paintings. [#permalink]  10 Apr 2019, 06:19
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Carcass wrote:
At an art gallery, there are 7 spaces to hang 14 paintings. The 14 paintings to choose from are: three by Cezanne, three by Monet, one by Renoir, two by Van Gogh, two by Pissarro, two by Degas, and one by Manet.

The even-numbered spaces must hold either a Cezanne or a Monet. The first space must hold a Renoir. The fifth space must hold a Manet, and the remaining spaces must hold either a Van Gogh, a Pissarro, or a Degas.

How many unique arrangements of paintings are possible?

A. 28

B. 72

C. 1200

D. 3600

E. 7776

Take the task of seating the 6 students and break it into stages.

Stage 1: Select a painting for space #1
This must be a Renoir.
There is 1 painting by Renoir. So, we can complete this stage in 1 way

Stage 2: Select a painting for space #2
This must be either a Cezanne or a Monet
There are 6 such paintings to choose from. So we can complete this stage in 6 ways

Stage 3: Select a painting for space #4
This must be either a Cezanne or a Monet
There are 5 paintings REMAINING to choose from. So we can complete this stage in 5 ways

Stage 4: Select a painting for space #6
This must be either a Cezanne or a Monet
There are 4 paintings REMAINING to choose from. So we can complete this stage in 4 ways

Stage 5: Select a painting for space #5
This must be a Manet
There is 1 painting by Manet. So we can complete this stage in 1 way

Stage 6: Select a painting for space #3
This must be a Van Gogh, a Pissarro, or a Degas.
There are 6 such paintings to choose from. So we can complete this stage in 6 ways

Stage 7: Select a painting for space #7
This must be a Van Gogh, a Pissarro, or a Degas.
There are 5 such paintings REMAINING to choose from. So we can complete this stage in 5 ways

By the Fundamental Counting Principle (FCP), we can complete all 7 stages (and thus fill all 7 spaces) in (1)(6)(5)(4)(1)(6)(5) ways (= 3600 ways)

Note: the FCP can be used to solve the MAJORITY of counting questions on the GRE. So, be sure to learn it.

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Re: At an art gallery, there are 7 spaces to hang 14 paintings.   [#permalink] 10 Apr 2019, 06:19
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