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

Question

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.)

A. 360
B. 625
C. 1024
D. 1296
 E. 6720

Bunuel · Accepted Answer

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:
https://gmatclub.com/forum/how-many-pos ... 85291.html
https://gmatclub.com/forum/combinations ... 08739.html
https://gmatclub.com/forum/solve-these- ... 98701.html
https://gmatclub.com/forum/voucher-98225.html

Theory on permutation and combinations:
https://gmatclub.com/forum/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.

Bunuel · Answer

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: https://gmatclub.com/forum/5-rings-on-4 ... er#p649557

docabuzar · Answer

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

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

Bunuel · Answer

There are 3 examples, the second one is the same as your question above and there is no contradiction between the solutions.

docabuzar · Answer

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!

docabuzar · Answer

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.

Bunuel · Answer

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.

email2vm · Answer

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.

theGame001 · Answer

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

dabaobao · Answer

Official Solution  
Credit: Veritas Prep

The first ring can be worn in 4 ways i.e. in any one of the four fingers. The second ring can be worn in 5 ways (it can go on any one of the four fingers and it can also go below the first ring so there are 5 distinct places for the second ring). The third ring can be worn in 6 ways (any one of the four fingers or below the second ring or below the first ring). The fourth ring can be worn in 7 ways (any one of the four fingers or below the third ring or below the second ring or below the first ring). The fifth ring can be worn in 8 ways (any one of the four fingers or below the fourth ring or below the third ring or below the second ring or below the first ring).

Total number of ways in which 5 different rings can be worn in 4 particular fingers = 4*5*6*7*8.

Ishtiaqur · Answer

Why can't we count the fingers instead. As in;finger 1 can have 0-5 rings on it. Hence 6 possibilities. The same goes for each of the other 4 fingers hence ans should be 6^4?

Posted from my mobile device

imslogic · Answer

In the scenario {2-2-1-0}, why is the 1 different fruit selected first and not the majority number as is the case with the other scenarios? Thank you for your review.

MBAHOUSE · Answer

To determine the number of ways 5 different rings can be worn on four particular fingers, we can use the concept of permutations with repetition. Since the rings are distinct and can be worn on the same finger, repetition is allowed.

Let's break down the problem:

1. Finger 1: The first finger can have 0 to 5 rings.

2. Finger 2: The second finger can have 0 to 5 rings.

3. Finger 3: The third finger can have 0 to 5 rings.

4. Finger 4: The fourth finger can have 0 to 5 rings.

For each finger, there are 6 options (0 rings, 1 ring, 2 rings, 3 rings, 4 rings, or 5 rings).

To find the total number of ways, we multiply the number of options for each finger:

Total ways = Options for Finger 1 × Options for Finger 2 × Options for Finger 3 × Options for Finger 4
Total ways = 6 × 6 × 6 × 6
Total ways = 1296

There are 1296 different ways to wear 5 distinct rings on four particular fingers, allowing some fingers to have more than one ring and others to have no rings.

Paras96 · Answer

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.)

No. of ways 1st ring can be worn = 4
No. of ways 2nd ring can be worn = 4
No. of ways 3rd ring can be worn = 4
.
No. of ways 5th ring can be worn = 4

Therefore total ways 5 different rings be worn in four particular fingers = 4*4*4*4*4 = 4^5
 
Hence Answer

adityaprateek15 · Answer

Bunuel  Could you please check why this question has no options available and update?

Bunuel · Answer

Added the options. Thank you!

Legallyblond · Answer

DISTRIBUTION OF THINGS INTO GROUPS   
 1. The number of ways in which mn different items can be divided equally into m groups, each containing n objects and the order of the groups is not important is  
  (mn)!(n!)m∗m!(mn)!(n!)m∗m!  
 .

2. The number of ways in which mn different items can be divided equally into m groups, each containing n objects and the order of the groups is important is  
  (mn)!(n!)m(mn)!(n!)m

3. The number of ways in which (m+n+p) things can be divided into three different groups of m,n, and p things respectively is  
  (m+n+p)!m!n!p!(m+n+p)!m!n!p!

4. The required number of ways of dividing 3n things into three groups of n each  
  (3n)!3!∗(n!)3(3n)!3!∗(n!)3

When the order of groups has importance then the required number of ways =  
  (3n)!(n!)3(3n)!(n!)3  
 . can we use this portion here? can you please explain what do we exactly do when different things comes into picture? i know ow to tackle identical things

mkeshri185 · Answer

We have 4 different fingers and we have 5 different rings and each ring can go to 4 finger so it should be 4^5. What's the catch here?

Legallyblond · Answer

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); when you are saying this are you talking about the arrnagement of the ring... like arent you double counting so if first ring goes in first finger and then the second ring can go in the first finger too which is counted in 4 ways then how 5? are you talking about arrangement?

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

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