Jean is arranging paint brushes used by artists for an impressionist's

Question

Jean is arranging paint brushes used by artists for an impressionist's exhibit. He has 4 from Van Gogh, 5 from Monet, 3 from Manet, 4 from Degas, and 2 from Toulouse-Lautrec to choose from. All brushes from the same artist are considered identical for ordering. Jean wants to group all the brushes from Van Gogh together. How many ways can he arrange the paint brushes by the artist that used them?

A) \(5!\)

B) \(15!\)

C) \(15!/4!\)

D) \(\frac{11!}{5! · 3! · 4! · 2!}\)

E) \(\frac{15!}{5! · 3! · 4! · 2!}\)

Source: Bloomberg Prep

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Official Answer

E

A)  5!

B)  15!

C)  15!/4!

D)  \frac{11!}{5! · 3! · 4! · 2!}

E)  \frac{15!}{5! · 3! · 4! · 2!}

Source: Bloomberg Prep

CrackverbalGMAT · Answer

Arrangement for something which is repeated

If n things are there where one thing is repeated x times, another is repeated y times and so on, then the total number of arrangements is = n! / (x! * y! * ...).

Since the Van Goghs have to be together, we consider them as 1 unit.

We now have 1 unit of Van Gogh + 5 Monet + 3 Manet + 4 Degas + 2 Toulouse-Lautre = 15

These can be arranged as  \frac{15!}{5! \space 3! \space 4! \space 2!}  ways

The Van Goghs can be arranged as  \frac{4!}{4!} = 1  way

Therefore the total arrangements =  \frac{15!}{5! \space 3! \space 4! \space 2!} * 1 = \frac{15!}{5! \space 3! \space 4! \space 2!}

Option E

Arun Kumar
Let us consider the 4 Van Goghs as 1 unit.

MathRevolution · Answer

Paintbrush: Van Gogh(V = 4), Monet(Mo = 5), Manet( Ma = 3), Degas(D = 4), and Toulouse-Lautrec( TL = 2)

Total brush: 4 + 5 + 3 + 4 + 2 = 18

Jean wants to group all the brushes from Van Gogh together: 14 brushes (V, V, V, V)  .

We have a total of 15 brushes and we can arrange those 15 brushes in 15! ways. 'V = 4' brushes can be arranged in 4! ways.

Therefore, ways to arrange: 15! * 4!

All brushes from the same artist are considered identical for ordering: We have to divide the result   by repetition

=>  \frac{(15! * 4!) }{ (4! * 5! * 3! * 4! * 2!)} = \frac{(15!) }{ (5! * 3! * 4! * 2!)}

Answer E

IanStewart · Answer

It's not clear we should use the Van Gogh or Toulouse-Lautrec brushes at all if it's an Impressionist exhibit, but art history aside, it's also not clear what constitutes an "arrangement" of paint brushes. Will they be set out in a row in a display case? Can they be in three rows, or in a circle? There's no way to answer the question without knowing what an "arrangement" is.

Assuming they're in a single row, then if all Van Gogh brushes go together, they might as well be one brush. Since the brushes from each individual artist are indistinguishable, we're then just making a "word" using the 15 'letters' V, Mo, Mo, Mo, Mo, Mo, Ma, Ma, Ma, D, D, D, D, TL, TL. If the letters were distinct, we'd have 15! orders, but we divide by 5! because the order of the 5 Mo's doesn't matter, by 3! because the order of the 3 Ma's doesn't matter, and so on, for an answer of 15! / (5!*3!*4!*2!)

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Jean is arranging paint brushes used by artists for an impressionist's

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