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Director  Joined: 22 Nov 2007
Posts: 820
A nickel, a dime, and 2 identical quarters are arranged  [#permalink]

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4
14 00:00

Difficulty:   55% (hard)

Question Stats: 57% (01:34) correct 43% (01:40) wrong based on 467 sessions

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A nickel, a dime, and 2 identical quarters are arranged along a side of a table. If the quarters and the dime have to face heads up, while the nickel can face either heads up or tails up, how many different arrangements of coins are possible?
A. 12
B. 24
C. 48
D. 72
E. 96

Originally posted by marcodonzelli on 31 Jan 2008, 22:22.
Last edited by Bunuel on 05 Feb 2012, 19:44, edited 1 time in total.
Edited the question and added the OA
Math Expert V
Joined: 02 Sep 2009
Posts: 60594
Re: still permutations with repetitions  [#permalink]

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13
15
T740qc wrote:
can somone explain this please? 4!/2! to get to diff ways the coins can be placed taking into acct the repeat in quarters but i don't get the *2! part. Thanks!

Welcome to Gmat Club.

THEORY.
Permutations of $$n$$ things of which $$P_1$$ are alike of one kind, $$P_2$$ are alike of second kind, $$P_3$$ are alike of third kind ... $$P_r$$ are alike of $$r_{th}$$ kind such that: $$P_1+P_2+P_3+..+P_r=n$$ is:

$$\frac{n!}{P_1!*P_2!*P_3!*...*P_r!}$$.

For example number of permutation of the letters of the word "gmatclub" is $$8!$$ as there are 8 DISTINCT letters in this word.

Number of permutation of the letters of the word "google" is $$\frac{6!}{2!2!}$$, as there are 6 letters out of which "g" and "o" are represented twice.

Number of permutation of 9 balls out of which 4 are red, 3 green and 2 blue, would be $$\frac{9!}{4!3!2!}$$.

BACK TO THE ORIGINAL QUESTION:
A nickel, a dime, and 2 identical quarters are arranged along a side of a table. If the quarters and the dime have to face heads up, while the nickel can face either heads up or tails up, how many different arrangements of coins are possible?
A. 12
B. 24
C. 48
D. 72
E. 96

# of arrangements of a nickel, a dime, and 2 quarters, or arrangements of 4 letters NDQQ out of which 2 Q's appear twice will be $$\frac{4!}{2!}$$. Next, as nickel can face either heads up or tails up then we should multiple this number by 2, so finally we'll get: $$\frac{4!}{2!}*2=24$$.

Hope it's clear.
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Intern  Joined: 05 Aug 2011
Posts: 6
Re: still permutations with repetitions  [#permalink]

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can somone explain this please? 4!/2! to get to diff ways the coins can be placed taking into acct the repeat in quarters but i don't get the *2! part. Thanks!
Manager  B
Joined: 12 Oct 2012
Posts: 105
WE: General Management (Other)
Re: still permutations with repetitions  [#permalink]

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Bunuel wrote:
T740qc wrote:
can somone explain this please? 4!/2! to get to diff ways the coins can be placed taking into acct the repeat in quarters but i don't get the *2! part. Thanks!

Welcome to Gmat Club.

THEORY.
Permutations of $$n$$ things of which $$P_1$$ are alike of one kind, $$P_2$$ are alike of second kind, $$P_3$$ are alike of third kind ... $$P_r$$ are alike of $$r_{th}$$ kind such that: $$P_1+P_2+P_3+..+P_r=n$$ is:

$$\frac{n!}{P_1!*P_2!*P_3!*...*P_r!}$$.

For example number of permutation of the letters of the word "gmatclub" is $$8!$$ as there are 8 DISTINCT letters in this word.

Number of permutation of the letters of the word "google" is $$\frac{6!}{2!2!}$$, as there are 6 letters out of which "g" and "o" are represented twice.

Number of permutation of 9 balls out of which 4 are red, 3 green and 2 blue, would be $$\frac{9!}{4!3!2!}$$.

BACK TO THE ORIGINAL QUESTION:
A nickel, a dime, and 2 identical quarters are arranged along a side of a table. If the quarters and the dime have to face heads up, while the nickel can face either heads up or tails up, how many different arrangements of coins are possible?
A. 12
B. 24
C. 48
D. 72
E. 96

# of arrangements of a nickel, a dime, and 2 quarters, or arrangements of 4 letters NDQQ out of which 2 Q's appear twice will be $$\frac{4!}{2!}$$. Next, as nickel can face either heads up or tails up then we should multiple this number by 2, so finally we'll get: $$\frac{4!}{2!}*2=24$$.

Hope it's clear.

I completely understood the concept. But i am little doubtful about one more condition given " If the quarters and dime have to face heads up"
Do we assume that they are facing up or we should subtract the possibility of having tails up??
Math Expert V
Joined: 02 Sep 2009
Posts: 60594
Re: still permutations with repetitions  [#permalink]

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1
Bunuel wrote:
T740qc wrote:
can somone explain this please? 4!/2! to get to diff ways the coins can be placed taking into acct the repeat in quarters but i don't get the *2! part. Thanks!

Welcome to Gmat Club.

THEORY.
Permutations of $$n$$ things of which $$P_1$$ are alike of one kind, $$P_2$$ are alike of second kind, $$P_3$$ are alike of third kind ... $$P_r$$ are alike of $$r_{th}$$ kind such that: $$P_1+P_2+P_3+..+P_r=n$$ is:

$$\frac{n!}{P_1!*P_2!*P_3!*...*P_r!}$$.

For example number of permutation of the letters of the word "gmatclub" is $$8!$$ as there are 8 DISTINCT letters in this word.

Number of permutation of the letters of the word "google" is $$\frac{6!}{2!2!}$$, as there are 6 letters out of which "g" and "o" are represented twice.

Number of permutation of 9 balls out of which 4 are red, 3 green and 2 blue, would be $$\frac{9!}{4!3!2!}$$.

BACK TO THE ORIGINAL QUESTION:
A nickel, a dime, and 2 identical quarters are arranged along a side of a table. If the quarters and the dime have to face heads up, while the nickel can face either heads up or tails up, how many different arrangements of coins are possible?
A. 12
B. 24
C. 48
D. 72
E. 96

# of arrangements of a nickel, a dime, and 2 quarters, or arrangements of 4 letters NDQQ out of which 2 Q's appear twice will be $$\frac{4!}{2!}$$. Next, as nickel can face either heads up or tails up then we should multiple this number by 2, so finally we'll get: $$\frac{4!}{2!}*2=24$$.

Hope it's clear.

I completely understood the concept. But i am little doubtful about one more condition given " If the quarters and dime have to face heads up"
Do we assume that they are facing up or we should subtract the possibility of having tails up??

The nickel can face either heads up or tails up, thus we multiply the total # of ways in which we can arrange NDQQ by 2. The quarters and the dime have to face heads up, so only 1 choice for both of them, thus we don't need to multiply further.

Hope it's clear.
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Posts: 394
Concentration: Marketing, Finance
GPA: 3.23
Re: A nickel, a dime, and 2 identical quarters are arranged  [#permalink]

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1
marcodonzelli wrote:
A nickel, a dime, and 2 identical quarters are arranged along a side of a table. If the quarters and the dime have to face heads up, while the nickel can face either heads up or tails up, how many different arrangements of coins are possible?
A. 12
B. 24
C. 48
D. 72
E. 96

How many ways to arrange {N}{D}{Q}{Q} ? $$=\frac{4!}{2!} = 12$$

$$=12*1*1*1*(2)$$ Since there are two ways to arrange the nickel...

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Concentration: Finance, Economics
GMAT 1: 650 Q49 V30 GPA: 3.92
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Re: A nickel, a dime, and 2 identical quarters are arranged  [#permalink]

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1
wrote 4!/2! * 2
and rewritten it as 4*3/1*2 and forgot to put on top 2! because of this, I got 12 Intern  B
Joined: 11 Apr 2016
Posts: 10
Re: A nickel, a dime, and 2 identical quarters are arranged  [#permalink]

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Hi, shouldn't the answer be 48.
For the Nickel and the Dime, we can select 2 places out of 4 in 4C2 ways followed by the arrangement in 2! ways. The two quarters then can have 4 combinations (HH, HT, TH, TT).

Hence the answer = 4C2 * 2! * 4 = 48. Please suggest Bunuel.

Please ignore any typos as I am new to this forum.
Manager  Joined: 13 Apr 2016
Posts: 55
Location: India
GMAT 1: 640 Q50 V27
GPA: 3
WE: Operations (Hospitality and Tourism)
Re: A nickel, a dime, and 2 identical quarters are arranged  [#permalink]

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[quote="marcodonzelli"]A nickel, a dime, and 2 identical quarters are arranged along a side of a table. If the quarters and the dime have to face heads up, while the nickel can face either heads up or tails up, how many different arrangements of coins are possible?
A. 12
B. 24
C. 48
D. 72
E. 96

No of ways when nickel face head=4!/2!=12
no of ways when nickel face tail= 4!/2!=12
total no of ways =24

Math Expert V
Joined: 02 Aug 2009
Posts: 8341
Re: A nickel, a dime, and 2 identical quarters are arranged  [#permalink]

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2
Hi, shouldn't the answer be 48.
For the Nickel and the Dime, we can select 2 places out of 4 in 4C2 ways followed by the arrangement in 2! ways. The two quarters then can have 4 combinations (HH, HT, TH, TT).

Hence the answer = 4C2 * 2! * 4 = 48. Please suggest Bunuel.

Please ignore any typos as I am new to this forum.

Hi,
It says that quarters and dime have to be faced heads up..
ONLY Nickel can be either head or tail..
so Nickel can be placed in two ways..
ans 4C2*2!*2= 24..
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Intern  B
Joined: 11 Apr 2016
Posts: 10
Re: A nickel, a dime, and 2 identical quarters are arranged  [#permalink]

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chetan2u - yes, misread the question. thanks.
Intern  Joined: 10 Sep 2016
Posts: 15
Re: A nickel, a dime, and 2 identical quarters are arranged  [#permalink]

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This clears it up!! Thanks a lot!!!

Posted from my mobile device
Intern  B
Joined: 26 Feb 2017
Posts: 23
Re: A nickel, a dime, and 2 identical quarters are arranged  [#permalink]

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Can anyone give a complete explanation for this

Sent from my ONE A2003 using GMAT Club Forum mobile app
Math Expert V
Joined: 02 Sep 2009
Posts: 60594
Re: A nickel, a dime, and 2 identical quarters are arranged  [#permalink]

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ashisplb wrote:
Can anyone give a complete explanation for this

Sent from my ONE A2003 using GMAT Club Forum mobile app

There are several solutions given above. You should be more specific when asking a question.
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Re: A nickel, a dime, and 2 identical quarters are arranged  [#permalink]

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1
+ B = 24

Attachment: FullSizeRender (9).jpg [ 68.03 KiB | Viewed 8708 times ]
Intern  B
Joined: 26 Feb 2017
Posts: 23
Re: A nickel, a dime, and 2 identical quarters are arranged  [#permalink]

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Please let me know whether my approach is correct or not

For 2 identical quarters =4C2
For dime =2 heads up are available
Nickel =either heads up or tails up =2
Total=4C2*2*2=24

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Manager  B
Joined: 25 Nov 2019
Posts: 50
Re: A nickel, a dime, and 2 identical quarters are arranged  [#permalink]

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nickles in position 1 as heads or tails and only the dime changes or N(2)QQD, N(2)QDQ, N(2)DQQ = 6
nickles in position 2 as heads or tails and only the dime changes or QN(2)QD, NQ(2)DQ, ND(2)QQ = 6
nickles in position 3 as heads or tails and only the dime changes or QQN(2)D, QDN(2)Q, DQN(2)Q = 6
nickles in position 4 as heads or tails and only the dime changes or QQDN(2), QDQN(2), DQQN(2) = 6
final count 24 seems slow but once you see the pattern it's easy Re: A nickel, a dime, and 2 identical quarters are arranged   [#permalink] 10 Dec 2019, 08:33
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