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# Tom is arranging his marble collection in a collector's case. He has

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Math Expert
Joined: 02 Sep 2009
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Tom is arranging his marble collection in a collector's case. He has  [#permalink]

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25 Dec 2017, 01:52
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Difficulty:

95% (hard)

Question Stats:

42% (03:14) correct 58% (03:00) wrong based on 33 sessions

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Tom is arranging his marble collection in a collector's 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 at least have 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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Re: Tom is arranging his marble collection in a collector's case. He has  [#permalink]

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25 Dec 2017, 21:55
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Bunuel wrote:
Tom is arranging his marble collection in a collector's 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 at least have 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

hi..

so 5Cs, 5Ss and 3As..
atleast one of each C,S and A means TWO cases
I.. 3 of one kind and one each of other two
$$\frac{5!}{3!}=20$$
any one of 3 can be three, so 20*3=60
II... 2 of two and one of third
$$\frac{5!}{2!2!}=30$$
any one of 3 can be one, so 30*3=90

total 60+90=150

B
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Re: Tom is arranging his marble collection in a collector's case. He has  [#permalink]

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28 Dec 2017, 13:18
Could someone pls provide more comprehensive explanation?
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Joined: 20 Feb 2017
Posts: 88
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Tom is arranging his marble collection in a collector's case. He has  [#permalink]

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28 Dec 2017, 15:04
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krikre wrote:
Could someone pls provide more comprehensive explanation?

Though explanation provided by Chetan is good, I am adding more details to his solution.

Tom has 5 cat-eys (say C) , 5 sulphides (say S) and 3 agates (say A).
So Tom has 5C, 5S and 3A.

The case can fit 5 marbles and please note the question is asking for the arrangement so the order of marbles in Case is also important.
As per the question. We must have 1C, 1S and 1A.
so for remaining 2 spots -
We can Choose 2 of one kind or 1 of two kinds.

1. If we choose 2 of one kind -
We can choose 2 of one kind in 3 ways ( Either 2C or 2A or 2C)
So in total in case, we will have 3 of one kind and 2 other kinds of marble
There is a formula to solve these kinds of arrangement but in case somebody doesn't recall -
We can arrange the 1 kind in 5 ways and second kind by 4. We have 3 identical marble left and 3 spots to fill so the order is not important now.
So ways = 5X4 = 20
so total ways as we can select first 2 marble in 3 ways
= 5X4X3 = 60

2. If we choose 2 kinds of marble to fill 2 empty spaces -
Now we can choose marble in 3 ways ( 1A and 1C, 1A and 1S, 1S and 1C).
So in case, we will 2-2 marbles of 2 kinds and 1 marble of one kind.
Lets' assume we have 2A, 2S and 1C.
for 1C, we can select spots in 5 ways.
for 2As, we can select spots in 6 ways (Select 2 out of 4, order not important).
for remaining 2S, we are left with 2 spots only after fixing 1C and 2As, so only one way.
So ways = 6X5 = 30
so total ways as we can select first 2 marble in 3 ways
= 30X3 = 90.

Total ways = Case 1 + Case 2 = 60 + 90 = 150

Note - It may be easier to solve with the formulas if you are aware of them.
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Re: Tom is arranging his marble collection in a collector's case. He has  [#permalink]

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28 Dec 2017, 16:55
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krikre wrote:
Could someone pls provide more comprehensive explanation?

Intern
Joined: 30 Jul 2017
Posts: 20
Re: Tom is arranging his marble collection in a collector's case. He has  [#permalink]

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30 Dec 2017, 15:57
Hello Lostin and Turkish, thank you for your responses.

Lostin could you kindly let me know what formulas you mean ?

Thank you.
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Re: Tom is arranging his marble collection in a collector's case. He has  [#permalink]

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30 Dec 2017, 16:40
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krikre wrote:
Hello Lostin and Turkish, thank you for your responses.

Lostin could you kindly let me know what formulas you mean ?

Thank you.

Chetan has used the same formulas for solutions, very handy and fast in case of big numbers.
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Re: Tom is arranging his marble collection in a collector's case. He has  [#permalink]

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30 Dec 2017, 17:03
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05 Jul 2019, 09:41
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Re: Tom is arranging his marble collection in a collector's case. He has   [#permalink] 05 Jul 2019, 09:41
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