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Example 3 from Algebra and Trigonometry, 13.5 Counting Principles
Finding the Number of Permutations Using the Multiplication Principle
At a swimming competition, nine swimmers compete in a race.
- ⓐHow many ways can they place first, second, and third?
- ⓑHow many ways can they place first, second, and third if a swimmer named Ariel wins first place? (Assume there is only one contestant named Ariel.)
- ⓒHow many ways can all nine swimmers line up for a photo?
Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.
Solution
- ⓐDraw lines for each place.
There are 9 options for first place. Once someone has won first place, there are 8 remaining options for second place. Once first and second place have been won, there are 7 remaining options for third place.
Multiply to find that there are 504 ways for the swimmers to place.
- ⓑDraw lines for describing each place.
We know Ariel must win first place, so there is only 1 option for first place. There are 8 remaining options for second place, and then 7 remaining options for third place.
Multiply to find that there are 56 ways for the swimmers to place if Ariel wins first.
- ⓒ
Draw lines for describing each place in the photo.
There are 9 choices for the first spot, then 8 for the second, 7 for the third, 6 for the fourth, and so on until only 1 person remains for the last spot.
There are 362,880 possible permutations for the swimmers to line up.
- ⓐDraw lines for each place.
Your turn
The book's Try It right after this example: same idea, new numbers. Only the answer is given.
How many ways can the family line up for the portrait?
How did it go?