# SAT Math: Arithmetic

## Counting Methods

At the 20th checkpoint in the Iditarod Trail Sled Dog Race, 5 mushers are considered to have an equal chance of winning the race.  If these 5 mushers each finish one right after the other, how many different ways are there for these 5 to finish the race?  Assume that none of the other participants in the race will place within the top 5 finishers.

A.  5

B.   15

C.  25

D.  120

E.  625

## Knowsys Method

Read the question carefully.   This problem has more numbers than you need worry about.  The 20th checkpoint is irrelevant, and the 5 is repeated a lot of times.  Focus on the relevant information by identifying the bottom line.

Identify the bottom line.  How many ways can 5 mushers finish a race?

Assess your options.  You could assign each musher a name or number and rearrange them as many ways as you can.  Using this method you are likely to miss some combinations and waste a lot of time.  Instead, use the Knowsys permutation mind method.

Attack the problem.  Permutation problems ask now many ways something can happen.  Each time you select a member from the group, the remaining number of possible choices decreases by one.  For example, if there are 5 mushers who could get first place and one of them gets first place, then there are only 4 mushers remaining who could be in second place.

The Knowsys mind method is to draw out enough lines to cover the scenario and then fill those lines with the number of possibilities that could go in each line.  When you have finished, you will multiply the numbers together.

For this problem there are 5 different finishing places that you are worried about:

__ * __ * __ * __ * __

Now each of the 5 mushers has an equal chance of getting first place.  Put a 5 on the first line.

* __ * __ * __ * __

If one musher has already finished first, then that only leaves 4 possibilities for second place.   If two mushers have finished first and second, then there are only 3 possible mushers to finish in third place.  Continue this pattern.

5 * 4 * 3 * 2 * 1

Now multiply those numbers together!  (If you have covered factorials in school, you will recognize this as 5!, which you may have memorized.)  5 x 4 x 3 x 2 x 1 = 120.

Loop back.  Checking your answer against your bottom line will show you that you found all of the possible ways for 5 mushers to finish a race.  Select your answer.