Maude wrote:
Hello
Please need help to understand the following
6 persons are going to theater and will sit next to each other in 6 adjacent seats
But Martia and Jan can not sit next to each other .In how many Arrangement can this be done
I understood that the restriction must be deal first
by finding the number of way the restriction happen and remove from the total number of way to arrange the n !
It is 2 ! for arrangement and 4 ! for the remaining 4 people
but what I don t understand is why it is time by 5 as the OA gives
I saw some other type like that
For instance digit 1,2,3,4,5 IF EACH DIGIT is used only once how many ways can each digit be arranged such 2 and 4 are not adjacent -
In this case the restriction is 2!x4! not multiplied by anything else
Can anyone explain me why
thanks for your Time
regards
Hi, and welcome to Gmat Club! Below is the solution for your problem.
You are right saying that probably the best way to deal with the questions like this is to count total # of arrangements and then subtract # of arrangements for which opposite of restriction occur. But the way you are calculating the later is not correct.
Total # of arrangements of 6 people (let's say A, B, C, D, E, F) is \(6!\).
# of arrangement for which 2 particular persons (let's say A and B)
are adjacent can be calculated as follows: consider these two persons as one unit like {AB}. We would have total 5 units: {AB}{C}{D}{E}{F} - # of arrangement of them 5!, # of arrangements of A and B within their unit is 2!, hence total # of arrangement when A and B are adjacent is \(5!*2!\).
# of arrangement when A and B are not adjacent is \(6!-5!*2!\).
In your example about 5 digits the answer would be:
Total # of arrangements of 5 distinct digits is \(5!\).
# of arrangement for which 2 digits 2 and 4
are adjacent is: consider these two digits as one unit like {24}. We would have total 4 units: {24}{1}{3}{5} - # of arrangement of them 4!, # of arrangements of 2 and 4 within their unit is 2!, hence total # of arrangement when 2 and 4 are adjacent is \(4!*2!\).
# of arrangement when 2 and 4 are not adjacent is \(5!-4!*2!\).
Hope it helps.
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