Problem Solving Competition

First Problem

How Many Arrangements?

A car holds 6 people (including the driver), 3 in the front seat and 3 in the back seat. How many different seating arrangements of the 6 people are possible if one person refuses to sit in the front and one different person refuses to sit in the back? Assume all 6 are licensed drivers.

Solution

If we look at the options of seating arrangements for each person, we get:
The person who won't ist in the back has 3 possible choices.
The person who won't sit in the front has 3 possible choices.
Then there are 4 people left with 4 choices, so there are 4! possiblities (because there are 4 possibilities for the first of them, then 3 possibilities for the second, then 2 possibilities for the third, and only 1 possibility for the last person, so we have 4*3*2*1 = 4!).
Therefore, 3*3*4! = 3*3*4*3*2*1 = 216 possiblities.