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Question on combinations and permutations [#permalink]

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11 Nov 2009, 10:27

So if you have 6 people going to a movie and 6 seats next to each other for them to sit, but 2 of the people, Ron and Todd, will not sit next to each other you have 6! - (5! + 5!) ways to sit everybody in this mannor. This is the total number of ways to sit 6 people, minus each of the possible ways that the two people, Ron and Todd, can sit next to each other.

First question: Is there a better strategy to solve this type of question without addition/subtraction?

Now although this may or may not be something the GMAT will ask, I am curious to the following. Lets say that of the 6 people, the two that wont sit next to each other, Ron and Todd, have decided that they WILL sit with EXACTLY 1 person in between them. How many different ways can the 6 people sit while having Ron and Todd separated by exactly 1 person?

The math/strategy behind the second part would really be appreciated.

Re: Question on combinations and permutations [#permalink]

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17 Nov 2009, 21:29

benjiboo wrote:

So if you have 6 people going to a movie and 6 seats next to each other for them to sit, but 2 of the people, Ron and Todd, will not sit next to each other you have 6! - (5! + 5!) ways to sit everybody in this mannor. This is the total number of ways to sit 6 people, minus each of the possible ways that the two people, Ron and Todd, can sit next to each other.

First question: Is there a better strategy to solve this type of question without addition/subtraction?

Now although this may or may not be something the GMAT will ask, I am curious to the following. Lets say that of the 6 people, the two that wont sit next to each other, Ron and Todd, have decided that they WILL sit with EXACTLY 1 person in between them. How many different ways can the 6 people sit while having Ron and Todd separated by exactly 1 person?

The math/strategy behind the second part would really be appreciated.

Re: Question on combinations and permutations [#permalink]

Show Tags

17 Nov 2009, 21:50

swatirpr wrote:

benjiboo wrote:

So if you have 6 people going to a movie and 6 seats next to each other for them to sit, but 2 of the people, Ron and Todd, will not sit next to each other you have 6! - (5! + 5!) ways to sit everybody in this mannor. This is the total number of ways to sit 6 people, minus each of the possible ways that the two people, Ron and Todd, can sit next to each other.

First question: Is there a better strategy to solve this type of question without addition/subtraction?

Now although this may or may not be something the GMAT will ask, I am curious to the following. Lets say that of the 6 people, the two that wont sit next to each other, Ron and Todd, have decided that they WILL sit with EXACTLY 1 person in between them. How many different ways can the 6 people sit while having Ron and Todd separated by exactly 1 person?

The math/strategy behind the second part would really be appreciated.

Re: Question on combinations and permutations [#permalink]

Show Tags

18 Nov 2009, 05:59

1

This post received KUDOS

benjiboo wrote:

swatirpr wrote:

benjiboo wrote:

So if you have 6 people going to a movie and 6 seats next to each other for them to sit, but 2 of the people, Ron and Todd, will not sit next to each other you have 6! - (5! + 5!) ways to sit everybody in this mannor. This is the total number of ways to sit 6 people, minus each of the possible ways that the two people, Ron and Todd, can sit next to each other.

First question: Is there a better strategy to solve this type of question without addition/subtraction?

Now although this may or may not be something the GMAT will ask, I am curious to the following. Lets say that of the 6 people, the two that wont sit next to each other, Ron and Todd, have decided that they WILL sit with EXACTLY 1 person in between them. How many different ways can the 6 people sit while having Ron and Todd separated by exactly 1 person?

The math/strategy behind the second part would really be appreciated.

condition - Ron and Todd separated by exactly 1 person

So If R @ 1 then T @ 3 - 1st way If T @ 1 then R @ 3 - 2nd way So R n T can sit 2 ways For remaining Seats 2, 4, 5, 6, 4 people can seat 4*3*2*1 ways Total for this seating arrangement 2*4*3*2*1 =48

OR

If R @ 2 then T @ 4 - 1st way If T @ 2 then R @ 4 - 2nd way So R n T can sit 2 ways For remaining Seats 1, 3, 5, 6, 4 people can seat 4*3*2*1 ways Total for this seating arrangement 2*4*3*2*1 =48

OR

If R @ 3 then T @ 5 - 1st way If T @ 3 then R @ 5 - 2nd way So R n T can sit 2 ways For remaining Seats 1, 2, 4, 6, 4 people can seat 4*3*2*1 ways Total for this seating arrangement 2*4*3*2*1 =48

Re: Question on combinations and permutations [#permalink]

Show Tags

21 Nov 2009, 18:12

Expert's post

benjiboo wrote:

So if you have 6 people going to a movie and 6 seats next to each other for them to sit, but 2 of the people, Ron and Todd, will not sit next to each other you have 6! - (5! + 5!) ways to sit everybody in this mannor. This is the total number of ways to sit 6 people, minus each of the possible ways that the two people, Ron and Todd, can sit next to each other.

First question: Is there a better strategy to solve this type of question without addition/subtraction?

Now although this may or may not be something the GMAT will ask, I am curious to the following. Lets say that of the 6 people, the two that wont sit next to each other, Ron and Todd, have decided that they WILL sit with EXACTLY 1 person in between them. How many different ways can the 6 people sit while having Ron and Todd separated by exactly 1 person?

The math/strategy behind the second part would really be appreciated.

Question #1: We have A, B, C, D, E and F. A and B don't want to sit together.

Let's count the # of ways when they sit together: glue them so that we have one unit from them {AB}. We'll have total of 5 units - {AB}{C}{D}{E}{F}. # of arrangements =5!. But we can fix {AB} as {BA} too so, 2*5!.

Total # of ways of arrangement of {A}{B}{C}{D}{E}{F}=6!.

# of arrangements when A and B will not sit together=6!-2*5!.

Question #2: We have A, B, C, D, E and F. We want A and B to sit so that any from C, D, E and F to be between them.

Again we can fix A and B, and any X between them: so we get 4 units: {ACB}{D}{E}{F}. # of combinations 4!. {ACB} also can be {BCA}, so 2*4!. But between A and B we can place any from the four not only C so 4*2*4!.

Re: Question on combinations and permutations [#permalink]

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22 Nov 2009, 00:52

This question cannot be done without addition/subtraction.

Reason being that there are 2 scenarios and each has their respective P&C. One scenario is for Ron/Todd to be seated at the first seat. The other scenario is when Ron/Todd are not sitting at the first seat.

Re: Question on combinations and permutations [#permalink]

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05 Jul 2015, 20:49

benjiboo wrote:

So if you have 6 people going to a movie and 6 seats next to each other for them to sit, but 2 of the people, Ron and Todd, will not sit next to each other you have 6! - (5! + 5!) ways to sit everybody in this mannor. This is the total number of ways to sit 6 people, minus each of the possible ways that the two people, Ron and Todd, can sit next to each other.

First question: Is there a better strategy to solve this type of question without addition/subtraction?

Now although this may or may not be something the GMAT will ask, I am curious to the following. Lets say that of the 6 people, the two that wont sit next to each other, Ron and Todd, have decided that they WILL sit with EXACTLY 1 person in between them. How many different ways can the 6 people sit while having Ron and Todd separated by exactly 1 person?

The math/strategy behind the second part would really be appreciated.

Ans for First Que.: It is itself a better strategy......there is no need to go for alternate when time matters.

Ans for Second Que. (Part) :

Consider that Ron, Todd and one of their friend(say X) have a single big seat and now the total available seats are 04.

Now in this case total combination will be 4! where it doesn’t matter Ron, Todd and X occupy which sitting order on the single big seat (viz. X is in between Ron and Todd or not).

Now, in case of X sitting in between Ron and Todd, no. of combination = 2 * 4!

Since, this order can be made with total four friends (including X) the final combination will be= 4 * 2* 4! = 192.

gmatclubot

Re: Question on combinations and permutations
[#permalink]
05 Jul 2015, 20:49

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