Problem number 17 states, 5 cars participated in a race, starting in the order shown. Whenever a car overtook another car, a point was awarded. The cars reached the finish line in the following order. What is the smallest number of points, in total, that could have been awarded? So here we have our cars in their current order at the top, and their desired order at the bottom. In order to calculate how many cars must pass every other car, let us analyze each car individually, and see what cars are in front of it initially, and which ones must be behind it at the end. From this we will be able to determine how many cars must overtake another car. Let's start out at the beginning with car number 1. Car 1 has cars 3 and 5 in front of it initially, but has cars 3 and 5 behind it at the end of the race. This means car 1 must pass cars 3 and 5. Car 2 must pass cars 3, 4, and 5 for the same reasons. Car 3 ends up in the back, so it will pass no cars. It will only be passed by cars. Car 4 must pass car 5, because initially it is in front of it, but at the end it is behind it. Car 5 passes no other cars, because it is in front, and ends up near the back. So in total, 6 cars must be passed. Here I can show you how this might look. First car 4 passes car 5. Then car 2 passes cars 3, 4, and 5. And lastly car 1 passes cars 3 and 5. So we can see that this works. We were able to do this with only 6 overtakes, which means 6 points. So the question asked us, what is the smallest number of points in total that could have been awarded? The answer is 6, letter E.

car race

overtake points

minimum points

race positions

overtakes total