Inequalities and permutations problem

[oussamam452]

Hello Everybody! Can someone please take a look at this problem and let me know how they solved it? I am stomped.

Dimitry is in charge of arranging the seating for the panel of judges at a jambalaya cook-off. There will be x judges sitting in a single consisting of x chairs. If x is greater than 3 but no more than 6, which of the following could be the number of ways Dimitry can arrange the judges?

A) 6
B) 25
C) 120
D) 500
E) 840

[Sage Pearce-Higgins]

If you have p objects and arrange them in a line, then there are p! different ways to arrange them. Although it's worth remembering this, it's also worth understanding it. Take a small number of objects, say 3 or 4, and list all the different orders, such as ABC, ACB, etc. as an understanding exercise.

Applying this knowledge to the problem above, you can see that the number of judges must be 4 or 5, so that the answer will be 4! (24) or 5! (120). Answer: C

[oussamam452]

Thank you for the detailed explanation! That was my reasoning also! For some reason, however, the answer key picked A. It could not have been 3! = 6...I suspected it to be wrong and that is why I posted this to double check...

[Sage Pearce-Higgins]

No problem. I'm glad that it makes sense now.