3 people, who go to a movie and sit next to each other in 4 adjacent

Question

Hi,

I have quite a question on basic counting principle and would really appreciate help.
Say, there are 3 people, who go to a movie and sit next to each other in 4 adjacent seats in the front row of the theatre.

Then should I say that ways in which the seats can be filled are:
= 3*2*1

Or should I say that ways in which people can choose their seats are:
= 4*3*2

Please can you explain if the above understanding is correct.
If so, why the values are different?

jayanthjanardhan · Answer

It will be 4x3x2.
The first person can be made to sit in the 4 seats in 4 ways. Once that is filled, the second person can be made to sit in 3 ways and so on...

GMATinsight · Answer

Hi  Patronus

You need to understand the reasoning behind 3x2x1 or 4x3x2

the Explanation of 3x2x1 can be given in this way

4 Seats and 3 persons

The First Seat can be occupied in 3 ways
The Second Seat can be occupied in 2 ways
The Third Seat can be occupied in 1 ways
i.e. 3x2x1 ways

But did you realize that happens to forth seat???
In fact you didn't give the forth seat a chance to be occupied and Neither did you give the first or second or third seat to remain vacant which is another possibility
 That's  unjustified distribution of opportunity  and therefore this method is INCORRECT

the Explanation of 4x3x2 can be given in this way

4 Seats and 3 persons

The First Individual has options to pick any seat out of four vacant seats
The Second Individual has options to pick any seat out of remaining three vacant seats
The Third Individual has options to pick any seat out of remaining Two vacant seats
i.e. 4x3x2 ways

In this way three members can sit on any three of the 4 seats and hence every seat get equal opportunity to be occupied and remain Vacant
 That's  justified distribution of opportunity  and therefore this method is CORRECT

GMATinsight · Answer

Another Method to use here is the  Method of Selection

Please Note: Selection i.e. n C r is used where you have more options and you need lesser of them

e.g. You go to store to buy 2 shirts and he shows you five then the ways to select 2 shirts out of 5 is 5 C 2=10
e.g. You have 10 candidates in team and you can promote only 3 of them then the ways to select 3 out of 10 at random is 10 C 3=120
 e.g. You have 4 Seats but only 3 of them have to be occupied then the ways to select 3 out of 4 at random is 4 C 3=4

In this question you can select 3 seats out of 4 at random in  4 C 3=4  ways and you can arrange 3 persons on those selected chairs in 3! ways

Hence total ways = 4 C 3*3! = 4*6 = 24 ways  which is same as 4x3x2

I hope it helps!

EMPOWERgmatRichC · Answer

Hi Patronus,

In this type of situation, since there are no restrictions on who can sit in any given seat, what you really have to account for are 4 "entities" - the three people AND one empty seat...

Thus, there are 4 options for the first seat. Once you place an 'entity'...
then there are 3 options for the second seat. Once you place an 'entity'...
then there are 2 options for the third seat....
and just 1 option for the fourth seat....

Thus (4)(3)(2)(1) = 24 is the correct number of permutations.

GMAT assassins aren't born, they're made,
Rich

3 people, who go to a movie and sit next to each other in 4 adjacent

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