SAT Math: Arithmetic

Counting Methods

A total of 15 dogs are competing in the local Dachshund Dash to raise money for a charity.  If only the first, second, and third place finishers get a prize, how many distinct possibilities are there for the combination of dogs that could take first, second, and third place?

A.   3

B.  6

C.   15

D.   42   

E.  2, 730


Knowsys Method

Read the question carefully.  It is important to notice that there are 15 total dogs, but you are only worried about the first 3 finishers.

Identify the bottom line.  How many different combinations are possible for first, second, and third place?

Assess your options.  You could assign each dog 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, there are 15 dogs that could finish first, but if one dog has already finished first, then there are only 14 that could finish second.

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 3 different finishing places that you are worried about.  You do not care about the other dogs that do not get prizes.

__ * __ * __

Now each of the 15 dogs has an equal chance of getting first place.  Put a 15 on the first line. 

15 * __ * __

If one dog has already finished first, then that only leaves 14 possibilities for second place.   If two dogs have finished first and second, then there are only 13 possible dogs to finish in third place:

15 * 14 * 13

Now multiply those numbers together!  The answer is 2,730.

Loop back.  Checking your answer against your bottom line will show you that you found all of the possible combinations for the first three places.


The correct answer is (D).

This is a medium level question.


Want some help reviewing the math concepts you need to master?  Try out the Knowsys Pre-Algebra Flashcards, the Knowsys Algebra I Flashcards, and the Knowsys SAT & ACT Math Practice book.  

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