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In how many ways can 5 different rings be worn in four

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In how many ways can 5 different rings be worn in four [#permalink] New post 02 Feb 2012, 15:16
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In how many ways can 5 different rings be worn in four particular fingers? (Some fingers may get more than one ring and some may get no rings.)

Can somebody explain?
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Re: 5 rings in four fingers [#permalink] New post 02 Feb 2012, 15:23
Expert's post
docabuzar wrote:
In how many ways can 5 different rings be worn in four particular fingers? (Some fingers may get more than one ring and some may get no rings.)

Can somebody explain?


First ring can be worn in 4 ways (on any of the four fingers);
Second ring can be worn in 5 ways (as it can go on any of four fingers - 4 ways; plus it can go below the first one - 1);
Third ring can be worn in 6 ways (the same logic as for the second ring);
Fourth ring can be worn in 7 ways;
Fifth ring can be worn in 8 ways;

Total: 4*5*6*7*8=6720.

Check this for other cases of the same question: 5-rings-on-4-fingers-86111.html?hilit=rings%20finger#p649557
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Re: 5 rings in four fingers [#permalink] New post 02 Feb 2012, 15:44
I tried to understand the ring-finger arrangements issue form the link you posted, but the trail appears contradictory. Since that was in 2009, can you pls re-confirm were your Answers & Explanations correct in that trail.

5-rings-on-4-fingers-86111.html?hilit=rings%20finger#p649557
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Re: 5 rings in four fingers [#permalink] New post 02 Feb 2012, 15:51
Expert's post
docabuzar wrote:
I tried to understand the ring-finger arrangements issue form the link you posted, but the trail appears contradictory. Since that was in 2009, can you pls re-confirm were your Answers & Explanations correct in that trail.

5-rings-on-4-fingers-86111.html?hilit=rings%20finger#p649557


There are 3 examples, the second one is the same as your question above and there is no contradiction between the solutions.
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Re: In how many ways can 5 different rings be worn in four [#permalink] New post 02 Feb 2012, 16:28
Many Thnx.

2 similar Qs.

1. In how many ways can 5 Idnetical fruits be distributed in 4 identical baskets?

Fruits are identical but baskets are also identical so we cannot apply 8C3 from n+r-1 C r-1?
Can we say that to remove the duplications of baskets we should divide 8C3 by 4! ?

2. In how many ways can 5 different fruits be distributed in 4 identical baskets?
Is this like making 4 groups from 5 different fruits so = 5!
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Re: In how many ways can 5 different rings be worn in four [#permalink] New post 02 Feb 2012, 17:24
Expert's post
docabuzar wrote:
Many Thnx.

2 similar Qs.

1. In how many ways can 5 Idnetical fruits be distributed in 4 identical baskets?

Fruits are identical but baskets are also identical so we cannot apply 8C3 from n+r-1 C r-1?
Can we say that to remove the duplications of baskets we should divide 8C3 by 4! ?

2. In how many ways can 5 different fruits be distributed in 4 identical baskets?
Is this like making 4 groups from 5 different fruits so = 5!


I think you are overthinking the concept by introducing the "identical basket" situation. For the first case if, for example, there is no difference between the following scenarios: {5-0-0-0} and {0-5-0-0}, then you can manually write down all possible cases: {5-0-0-0}, {4-1-0-0}, {3-1-1-0}, {3-2-0-0}, {2-2-1-0}, {2-1-1-1}. So the answer for the first question as you stated would be 6.

Don't overcomplicate it: the GMAT combination/probability questions are fairly straightforward and no need to waste time on the problems you will never see on the test.

If the questions were:

1. In how many ways can 5 identical fruits be distributed in 4 different baskets?

Consider five stars and three bars: *****|||, where stars represent fruits and three bars (one less than the baskets) will help us to divide them among the baskets. How many permutations (arrangements) of these 5+3=8 symbols are there? Permutation of 8 symbols out of which 5 * and 3 | are identical is 8!/(5!3!). Now, each arrangement will mean different scenario for fruit distribution. For example: **|*|*|* will mean that the first basket gets 2 fruits, the second, third and fourth 1 fruit. Or: ***|*|| will mean that the first basket gets 3 fruits, the second 1, and third and fourth none, and so on.

Answer: 8!/(5!3!).

2. In how many ways can 5 different fruits be distributed in different 4 baskets?

This one is easier each of the 5 different fruits has 4 choices, so total # of distribution is 4*4*4*4*4=4^5.

Answer: 4^5.

Check similar questions:
how-many-positive-integers-less-than-10-000-are-there-in-85291.html
combinations-tough-108739.html
solve-these-gmat-question-98701.html
voucher-98225.html

Theory on permutation and combinations:
permutation-86687.html

Direct formula if needed:
The total number of ways of dividing n identical items among r persons, each one of whom, can receive 0,1,2 or more items is n+r-1C_{r-1}.

The total number of ways of dividing n identical items among r persons, each one of whom receives at least one item is n-1C_{r-1}.


Hope it helps.
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COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: In how many ways can 5 different rings be worn in four [#permalink] New post 02 Feb 2012, 17:46
Many Thanks for the clarification.

Both the original question in the post above with "Identical basket" & different basket were given in a post by veritasprep i think by Kbansal, I read & saved some time back.
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Re: In how many ways can 5 different rings be worn in four [#permalink] New post 02 Feb 2012, 18:18
Expert's post
docabuzar wrote:
Many Thanks for the clarification.

Both the original question in the post above with "Identical basket" & different basket were given in a post by veritasprep i think by Kbansal, I read & saved some time back.


OK. Below is the solution for your second question as it was originally stated. Notice that it's not that hard but it involves some lengthy calculations and that's the reason it's doubtful one encounter such problem on the test.


In how many ways can 5 different fruits be distributed in 4 identical baskets?

You can have 6 cases and each will have its own # of combinations:
{5-0-0-0} --> 1 way (as there is no difference in which identical basket all the fruits will go);
{4-1-0-0} --> C^4_5=5: choosing which 4 fruits out of 5 will be together in one basket;
{3-1-1-0} --> C^3_5=10: choosing which 3 fruits out of 5 will be together in one basket (it doesn't matter in which 2 baskets the remaining 2 fruit sill go);
{3-2-0-0} --> C^3_5=10: choosing which 3 fruits out of 5 will be together in one basket (it doesn't matter in which 1 baskets the remaining 2 fruit sill go);
{2-2-1-0} --> C^1_5*\frac{C^2_4}{2!}=15: choosing which single fruit goes to one basket and \frac{C^2_4}{2!} is the # of ways to split 4 different fruits in two baskets when the order of the baskets doesn't matter;
{2-1-1-1} --> C^2_5=10: choosing which 2 fruits out of 5 will be together in one basket;

Total: 1+5+10+10+15+10=51.
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COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: In how many ways can 5 different rings be worn in four [#permalink] New post 03 Aug 2013, 03:47
isn't the answer:

choose 4 rings out of 6 which means C(6,4) but I can choose to wear any ring in any finger so 4!

so answer should be C(6,4)* 4!

Please correct me if I am wrong.
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Re: In how many ways can 5 different rings be worn in four [#permalink] New post 29 Aug 2013, 21:35
Bunuel wrote:
docabuzar wrote:
Many Thnx.

2 similar Qs.

1. In how many ways can 5 Idnetical fruits be distributed in 4 identical baskets?

Fruits are identical but baskets are also identical so we cannot apply 8C3 from n+r-1 C r-1?
Can we say that to remove the duplications of baskets we should divide 8C3 by 4! ?

2. In how many ways can 5 different fruits be distributed in 4 identical baskets?
Is this like making 4 groups from 5 different fruits so = 5!


I think you are overthinking the concept by introducing the "identical basket" situation. For the first case if, for example, there is no difference between the following scenarios: {5-0-0-0} and {0-5-0-0}, then you can manually write down all possible cases: {5-0-0-0}, {4-1-0-0}, {3-1-1-0}, {3-2-0-0}, {2-2-1-0}, {2-1-1-1}. So the answer for the first question as you stated would be 6.

Don't overcomplicate it: the GMAT combination/probability questions are fairly straightforward and no need to waste time on the problems you will never see on the test.

If the questions were:

1. In how many ways can 5 identical fruits be distributed in 4 different baskets?

Consider five stars and three bars: *****|||, where stars represent fruits and three bars (one less than the baskets) will help us to divide them among the baskets. How many permutations (arrangements) of these 5+3=8 symbols are there? Permutation of 8 symbols out of which 5 * and 3 | are identical is 8!/(5!3!). Now, each arrangement will mean different scenario for fruit distribution. For example: **|*|*|* will mean that the first basket gets 2 fruits, the second, third and fourth 1 fruit. Or: ***|*|| will mean that the first basket gets 3 fruits, the second 1, and third and fourth none, and so on.


Answer: 8!/(5!3!).

2. In how many ways can 5 different fruits be distributed in different 4 baskets?

This one is easier each of the 5 different fruits has 4 choices, so total # of distribution is 4*4*4*4*4=4^5.

Answer: 4^5.

Check similar questions:
how-many-positive-integers-less-than-10-000-are-there-in-85291.html
combinations-tough-108739.html
solve-these-gmat-question-98701.html
voucher-98225.html

Theory on permutation and combinations:
permutation-86687.html

Direct formula if needed:
The total number of ways of dividing n identical items among r persons, each one of whom, can receive 0,1,2 or more items is n+r-1C_{r-1}.

The total number of ways of dividing n identical items among r persons, each one of whom receives at least one item is n-1C_{r-1}.


Hope it helps.


Hi,

Why cant we use the same logic as used in the ring problem? How does using identical and different change the solution? I mean these are not word problems where the combination has to be unique?

Thanks
Re: In how many ways can 5 different rings be worn in four   [#permalink] 29 Aug 2013, 21:35
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