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

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Bunuel
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Jean is arranging paint brushes used by artists for an impressionist's [#permalink] New post   20 Dec 2019, 01:19
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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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Re: Jean is arranging paint brushes used by artists for an impressionist's [#permalink] New post   20 Dec 2019, 02:10
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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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Explanation:

As all the brushes of Van Gogh brushes are together, we can count 1 brush

Van Gogh = 1
Monet = 5
Manet = 3
Degas = 4
Toulouse-Lautrec = 2

Total number of ways to arrange 15 brushes = (1 + 5 + 3 + 4 + 2)! = 15!

Since all brushes from the same artist are considered identical for ordering, we must take think into account.

Thus, the number of ways he can arrange the paint brushes by the artist = 15!/5!3!4!2!.

IMO-E

:please Please give kudos, If you find my explanation good enough :please
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Re: Jean is arranging paint brushes used by artists for an impressionist's [#permalink] New post   20 Dec 2019, 05:39
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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consider all vangough paint brushes as tied in band since they always have to be together
so total brushes are 1(all van gugh as a bunch)+ 5+3+4+2 =15!
and since each brush by a painter is similar
hence divide by similar groups
\(\frac{15!}{!5!3!4!2!}\) =\(\frac{15!}{5!3!4!2!}\)
thus answer E
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Re: Jean is arranging paint brushes used by artists for an impressionist's [#permalink] New post   22 Dec 2019, 20:49
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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] New post   25 Dec 2019, 11:11
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]
25 Dec 2019, 11:11
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