Tom is arranging his marble collection in a collectors case. : GMAT Problem Solving (PS)
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# Tom is arranging his marble collection in a collectors case.

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Tom is arranging his marble collection in a collectors case. [#permalink]

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10 Oct 2010, 02:08
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Tom is arranging his marble collection in a collectors case. He has five identical cat-eyes, five identical sulphides, and three identical agates. If he can fit exactly five marbles into the case and must have at least one of each type, how many different ways can he arrange the case?

(A) 120
(B) 150
(C) 420
(D) 1,260
(E) 1,680

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[Reveal] Spoiler: OA

Last edited by hemanthp on 10 Oct 2010, 02:36, edited 1 time in total.
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10 Oct 2010, 02:16
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hemanthp wrote:
Tom is arranging his marble collection in a collectors case. He has five identical cat-eyes, five identical sulphides, and three identical agates. If he can fit exactly five marbles into the case and must have at least one of each type, how many different ways can he arrange the case?

(A) 120
(B) 150
(C) 420
(D) 1,260
(E) 1,680

I dont agree with the OA. Lets discuss.

Given the constraints, there is 2 ways to choose the stones :

1) Take 2 each of 2 types and 1 of the third type
3 ways to choose the type for which 1 stone is to be taken
Ways to arrange = 5!/(2!x2!) = 30
Total = 30 x 3 = 90

2) Take 3 of 1 type, one each of the other types
3 ways to choose the type for which 3 stones are to be taken
Ways to arrange = 5!/3! = 20
Total = 20 x 3 = 60

So net number of ways = 150

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10 Oct 2010, 02:32
Damn... silly mistakes again. Thank you Shrouded. Feel embarrassed for making a mistake and posting it out thinking I was right. I feel I should delete the post.
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10 Oct 2010, 04:15
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hemanthp wrote:
Tom is arranging his marble collection in a collectors case. He has five identical cat-eyes, five identical sulphides, and three identical agates. If he can fit exactly five marbles into the case and must have at least one of each type, how many different ways can he arrange the case?

(A) 120
(B) 150
(C) 420
(D) 1,260
(E) 1,680

Don't forget KUDOS if you like this.

We have that 3 marbles CSA must be in the collection case as he must have at least one of each type.

Then five chosen marbles could be:

1. CSA-XX - meaning that other 2 marbles are the same (for example CSA-CC):
$$\frac{5!}{3!}*C^1_3=60$$ ($$C^1_3$$ choosing which marble out of the 3 will be X and $$\frac{5!}{3!}$$ arranging 5 marbles our of which 3 are identical).

2. CSA-XY - meaning that other 2 marbles are not the same (for example CSA-CS):
$$\frac{5!}{2!2!}*C^2_3=60$$ ($$C^2_3$$ choosing which 2 marbles out of 3 will be X and Y and $$\frac{5!}{2!2!}$$ arranging 5 marbles our of which 2 and other 2 are identical).

Total: $$60+90=150$$.

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10 Oct 2010, 11:14
Thanks for the question and Explanation. Kudos to both of you
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Re: Tom is arranging his marble collection in a collectors case. [#permalink]

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03 Dec 2013, 06:50
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Re: Tom is arranging his marble collection in a collectors case. [#permalink]

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04 Dec 2013, 15:32
Doesn't the order of the stones matter in this problem? It says "how many different ways could he arrange the case?".
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Re: Tom is arranging his marble collection in a collectors case. [#permalink]

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05 Dec 2013, 01:53
PKUTNEY wrote:
Doesn't the order of the stones matter in this problem? It says "how many different ways could he arrange the case?".

Yes, it does and each solution above talks about arrangement of the marbles after selecting. For example, check here: tom-is-arranging-his-marble-collection-in-a-collectors-case-102588.html#p797440

Hope it helps.
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Re: Tom is arranging his marble collection in a collectors case. [#permalink]

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10 Dec 2013, 09:31
Either we could have 2,2,1 combo or 3,1,1 combo. Each of these combos in turn have 3 different cases.
So, required answer is 3*(5!/3!) + 3*(5!/2!*2!) = 150.

So, the correct answer is B.
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Re: Tom is arranging his marble collection in a collectors case. [#permalink]

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02 Aug 2015, 19:24
Hello from the GMAT Club BumpBot!

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Re: Tom is arranging his marble collection in a collectors case. [#permalink]

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07 Sep 2016, 10:55
Hello from the GMAT Club BumpBot!

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Re: Tom is arranging his marble collection in a collectors case.   [#permalink] 07 Sep 2016, 10:55
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