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# A magical leather pouch contains 2 alabaster marbles, 3 cerulean marbl

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Math Expert
Joined: 02 Sep 2009
Posts: 54369
A magical leather pouch contains 2 alabaster marbles, 3 cerulean marbl  [#permalink]

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19 Jul 2017, 23:57
14
00:00

Difficulty:

95% (hard)

Question Stats:

38% (03:09) correct 62% (03:11) wrong based on 73 sessions

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A magical leather pouch contains 2 alabaster marbles, 3 cerulean marbles, and 5 magenta marbles. First, four marbles are removed at random, without replacement. Next, three more marbles are removed at random, without replacement. What is the probability that the remaining marbles are the same color?

(A) 1/7
(B) 1/8
(C) 1/120
(D) 7/120
(E) 11/120

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Senior Manager
Joined: 24 Apr 2016
Posts: 331
A magical leather pouch contains 2 alabaster marbles, 3 cerulean marbl  [#permalink]

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03 Aug 2017, 21:45
4
1
The last three marbles can be either cerulean marbles or magenta marbles.

So how if we reverse the selection, i.e First three marbles are selected without replacement, such that the first three marbles are same.

Case 1: First three marbles are Cerulean =$$\frac{3}{10} * \frac{2}{9} * \frac{1}{8} = \frac{6}{720}$$

Case 2: First three marbles are Magenta = $$\frac{5}{10} * \frac{4}{9} * \frac{3}{8} = \frac{60}{720}$$

Total Probability = $$\frac{6}{720} + \frac{60}{720} = \frac{66}{720} = \frac{11}{120}$$

##### General Discussion
Manager
Joined: 02 Nov 2015
Posts: 163
GMAT 1: 640 Q49 V29
Re: A magical leather pouch contains 2 alabaster marbles, 3 cerulean marbl  [#permalink]

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20 Jul 2017, 00:16
Total outcomes 10c4 X 6c3. This equals total of 4200 outcomes.
Then we will take out 4 out of 7 in 7c4 ways an then 3 out of 3 in 1 ways. Thus probability equals (7c4X 3c3) / 4200.
Thus 1/120. Therefor C is the answer.

Sent from my Lenovo TAB S8-50LC using GMAT Club Forum mobile app
Intern
Joined: 02 Jul 2017
Posts: 3
Re: A magical leather pouch contains 2 alabaster marbles, 3 cerulean marbl  [#permalink]

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03 Aug 2017, 17:44
2
Basically we are picking 7 marbles out of 10 and the number ways to do that is 10C7 = 120
Now basically for having same color of marbles we have 2 options, either all cerulean marbles or 3 of the magenta are remaining.

so 2 alabaster + 5 magenta or 3 cerulean + 2 alabaster + 2 magenta

2a* 5m is 1 combination = 1* 1 = 1
3c * 2a * 5C2 = 1 * 1* 10

So totally 11 out 120 = 11/120

(E)
Manager
Joined: 07 Jun 2017
Posts: 100
Re: A magical leather pouch contains 2 alabaster marbles, 3 cerulean marbl  [#permalink]

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03 Aug 2017, 21:17
kumarparitosh123 wrote:
Total outcomes 10c4 X 6c3. This equals total of 4200 outcomes.
Then we will take out 4 out of 7 in 7c4 ways an then 3 out of 3 in 1 ways. Thus probability equals (7c4X 3c3) / 4200.
Thus 1/120. Therefor C is the answer.

Sent from my Lenovo TAB S8-50LC using GMAT Club Forum mobile app

Dear,
Would you explain what is 7c4 and 3c3?
Thank you so much.
Manager
Joined: 07 Jun 2017
Posts: 100
Re: A magical leather pouch contains 2 alabaster marbles, 3 cerulean marbl  [#permalink]

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03 Aug 2017, 21:21
ganj28 wrote:
Basically we are picking 7 marbles out of 10 and the number ways to do that is 10C7 = 120
Now basically for having same color of marbles we have 2 options, either all cerulean marbles or 3 of the magenta are remaining.

so 2 alabaster + 5 magenta or 3 cerulean + 2 alabaster + 2 magenta

2a* 5m is 1 combination = 1* 1 = 1
3c * 2a * 5C2 = 1 * 1* 10

So totally 11 out 120 = 11/120

(E)

Dear,
How is 2a*5m =1*1?
3c*2a = 1*1?
I really like your approach.. but i don't understand this part..
please explain.. thank you so much!
Intern
Joined: 02 Jul 2017
Posts: 3
Re: A magical leather pouch contains 2 alabaster marbles, 3 cerulean marbl  [#permalink]

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04 Aug 2017, 10:25
pclawong wrote:
ganj28 wrote:
Basically we are picking 7 marbles out of 10 and the number ways to do that is 10C7 = 120
Now basically for having same color of marbles we have 2 options, either all cerulean marbles or 3 of the magenta are remaining.

so 2 alabaster + 5 magenta or 3 cerulean + 2 alabaster + 2 magenta

2a* 5m is 1 combination = 1* 1 = 1
3c * 2a * 5C2 = 1 * 1* 10

So totally 11 out 120 = 11/120

(E)

Dear,
How is 2a*5m =1*1?
3c*2a = 1*1?
I really like your approach.. but i don't understand this part..
please explain.. thank you so much!

Number of ways of selecting 2 alabaster marbles and 5 magenta marbles = 1
Number of ways of selecting 3 cerulean , 2alabaster and 2 out of 5 magenta = 1*1*5C2 = 10

Hope its clear.
Manager
Joined: 12 Sep 2016
Posts: 65
Location: India
GMAT 1: 700 Q50 V34
GPA: 3.15
A magical leather pouch contains 2 alabaster marbles, 3 cerulean marbl  [#permalink]

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05 Aug 2017, 10:05
A magical leather pouch contains 2 alabaster marbles, 3 cerulean marbles, and 5 magenta marbles. First, four marbles are removed at random, without replacement. Next, three more marbles are removed at random, without replacement. What is the probability that the remaining marbles are the same color?

(A) 1/7
(B) 1/8
(C) 1/120
(D) 7/120
(E) 11/120

Total marbles = 10

After removal of 4 and then 3 marbles , remaining marbles = 10-4-3 = 3

The only way 3 marbles can be of the same color is when the marbles are either cerulean or magenta.

This question can be reinterpreted as :
what is the probability of drawing 3 cerulean or 3 magenta marbles from 2 alabaster marbles, 3 cerulean marbles, and 5 magenta marbles.

No of ways of drawing 3 marbles out of 10 = 10C3 = 120

No of ways of drawing 3 Cerulean marbles out of 3 = 3C3 =1

No of ways of drawing 3 Magenta marbles out of 5 = 5C3 =10

Therefore probability = (10+1) / 120 = 11/120
Manager
Joined: 02 Nov 2015
Posts: 163
GMAT 1: 640 Q49 V29
Re: A magical leather pouch contains 2 alabaster marbles, 3 cerulean marbl  [#permalink]

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05 Aug 2017, 11:56
pclawong wrote:
kumarparitosh123 wrote:
Total outcomes 10c4 X 6c3. This equals total of 4200 outcomes.
Then we will take out 4 out of 7 in 7c4 ways an then 3 out of 3 in 1 ways. Thus probability equals (7c4X 3c3) / 4200.
Thus 1/120. Therefor C is the answer.

Sent from my Lenovo TAB S8-50LC using GMAT Club Forum mobile app

Dear,
Would you explain what is 7c4 and 3c3?
Thank you so much.

No my approach and solution for this problem are flawed.

Thanks for pointing it.

Regards

Sent from my Lenovo TAB S8-50LC using GMAT Club Forum mobile app
Manhattan Prep Instructor
Joined: 04 Dec 2015
Posts: 729
GMAT 1: 790 Q51 V49
GRE 1: Q170 V170
Re: A magical leather pouch contains 2 alabaster marbles, 3 cerulean marbl  [#permalink]

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05 Aug 2017, 14:29
1
The sequential removal of marbles is a total red herring and you can ignore it. (This happens sometimes in combinatorics problems! To spot cases like this, think about whether what the problem is telling you is actually different from the simpler case. Is removing three marbles then four marbles (without looking at them) different from removing 7 marbles? Nope - that's like saying that eating four cookies then three cookies right afterwards, is somehow different from eating seven cookies all in a row.)

Now let's build up the probability.

probability = good cases / total cases

A 'good' case is one where the three marbles remaining are all the same color. They can't all be alabaster, since there definitely aren't enough alabaster marbles. They could all be cerulean; since there are exactly three cerulean marbles, there's only one way for that to happen. They could also all be magenta. Suppose that the magenta marbles were labeled A, B, C, D, and E. How many possible cases are there where three magenta marbles are left over?

ABC
ABD
ABE
ACD
ACE
BCD
BCE
BDE
CDE

That's ten possibilities. So, there are eleven possible cases where the three marbles left over are all the same color.

I stopped at this point and picked the one answer choice that has 11 in the numerator.

That's what you should do on the GMAT! But if you're just curious about the math, here's where the 120 in the denominator comes from. How many total cases are there? That is, how many ways can 3 marbles be left over, whether they're all the same or not? That's '10 choose 3', or 10!/(7!*3!), which equals 120.
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Re: A magical leather pouch contains 2 alabaster marbles, 3 cerulean marbl   [#permalink] 05 Aug 2017, 14:29
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