Question 1138961: In how many ways can the eight members of a students’ council pose in a line for a yearbook photograph if the chair and vice-chair must be side by side?
This is how I think it should be solved:
Since the chair and vice chair have to be together, they count as 1 person. That means there are 7 people to arrange, so there are 7!= 5040 possible orders. However, since the chair and vice chair can be in 2 possible orders relative to each other (one on the right and the other on the left), that makes the total possible arrangements 5040x2= 10,080.
This is the site's explanation:
First find the number of arrangements in which the chair and vice-chair are together. Consider the chair and vice-chair as a unit. This pair can be arranged with the remaining six members in 6P6=720 ways. For each of these ways, the chair could be either on the left or the right of the vice-chair.
Therefore, there is a total of 2×720=1440 ways in which the chair and vice-chair are together.
Why did they do 6! and not 7! ?
Found 2 solutions by jim_thompson5910, ikleyn:
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
You have the correct line of thinking and the correct final answer.
You basically have seven letters
A,B,C,D,E,F,G
such that six of the letters are individual people while the seventh is composed of two people as one unit. For any permutation of the seven items, you can swap the two people sitting together, hence the "times 2" portion. Basically this is paraphrasing your awesome explanation you posted.
The line of thinking that you would go with 6! is not fully correct because it only considers arranging {A,B,C,D,E,F,G} such that one letter (say the letter A) stays in one spot and the other six permute or move around. You have to consider the letter A moving around as well to get the full picture. I can see why they went with 6! because 8-2 = 6 people are free to move around as opposed to the 2 individuals who must stay together. If it was a case where the chair and vice chair people must be in the first two slots, then the answer would be 2*6! = 2*720 = 1440; however this is not the case.
So once again, the answer of 10,080 is correct. Nice work on getting the correct answer. Also I appreciate you posting your thoughts and your attempts at the answer, rather than just post the problem all by itself.
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
Your answer and your solution are correct.
The other solution and the other answer are WRONG.
They are wrong, because they do not account for 7 different possible positions of this pair (chair, vice-chair) in the line.
It is why they lost the factor 7.
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