Week 1 Challenge Question 6

Problem:
Alicia, Ben, Chen and Diana are lining up to see a movie. Which one of these is true about the number of ways there are for Alicia to be first in line?Explanation:
By symmetry, it doesn't matter which person we fix the position of. The number of ways for Alicia to be first is the same as the number of ways for Ben to be first, or Chen to be first, or Diana to be first. Interestingly enough, the position which we give the person also doesn't matter. If Chen stands at the front of the line, then Alicia has \(3\) places she can stand, Ben has \(2\) places he can stand and Diana has \(1\) place she can stand, so the total number of ways they can line up is \(3 \times 2 \times 1=6.\) If Chen is fixed to be second in line, Alicia still has \(3\) places she can stand, Ben still has \(2\) places he can stand, and Diana still has \(1\) place to stand. Since the person and position which the person is fixed do not change the number of ways that the others can line up, then the number of ways for Alicia to be first is the same as the number of ways for Chen to be third, answer choice \(\fbox{1}.\)Isn't the number of ways to arrange them in a circle six, because \(\frac{4!}{4}=3!=6?\) (It's \(\frac{4!}{4}\) because there's \(4!\) ways to arrange them in a line and \(4\) ways to rotate their positions.)

@spaceblastxy1428 Yes, the number of ways to arrange four people in a circle would be six. You are correct. The question is asking for something different, though: the four people here are standing in a line. Thank you!

In the question it says that Alicia has to be first in line, so there are \(3!=6\) ways, which is equal to the ways they can be arranged in a circle. The question only asks you if the answers are the same.