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Re: Jean is arranging paint brushes used by artists for an impressionist's [#permalink]
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Bunuel wrote:
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) \(\frac{15!}{4!}\)

(D) \(\frac{11!}{5!3!4!2!}\)

(E) \(\frac{15!}{5!3!4!2!}\)


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The brushes can be arranged as follows:

[V-V-V-V]-MO-MO-MO-MO-MO-MA-MA-MA-D-D-D-D-T-T

Since the Van Gogh brushes must be together, we consider those as “one” brush, so we are actually ordering 15 items, not 18. Additionally, since the brushes from the other artists are indistinguishable, we divide by the number of those indistinguishable brushes. We use the indistinguishable permutations formula, and the total number of arrangements becomes

15! / (5!3!4!2!)

Answer: E
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Re: Jean is arranging paint brushes used by artists for an impressionist's [#permalink]
Bunuel wrote:
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) \(\frac{15!}{4!}\)

(D) \(\frac{11!}{5!3!4!2!}\)

(E) \(\frac{15!}{5!3!4!2!}\)

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Since 4 brushes of Van Gogh are arranged together all the times, they can be considered as one unit, the arrangement within which doesn't matter.
So virtually total of 15 brushes are there now where 5 from Monet, 3 from Manet, 4 from Degas, and 2 from Toulouse-Lautrec are to choose from.

Hence arrangements possible are = \(\frac{15!}{5!4!3!2!}\)

Answer D.
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Re: Jean is arranging paint brushes used by artists for an impressionist's [#permalink]
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