Here are some carefully chosen permutation word problems that will show you how to solve word problems involving permutations.
Use the permutation formula shown below when the order is important.
Let _{n}P_{r} be the number of permutations of n objects arranged r at a time.
_{n}P_{r} = n(n - 1)(n - 2)(n - 3) ...
n is the first factor
Stop when there are r factors
The permutations word problems will show you how to do the followings:
Word problem #1
Eight cars enter a race. The three fastest cars will be given first, second, and third places. How many arrangements of first, second, and third places are possible with eight cars?
Solution
Here, the order does matter since they are not just picking any 3 cars regardless of how fast they drive. They are picking the three fastest cars to give them first, second, third places.
Evaluate _{n}P_{r} with n = 8 and r = 3
_{8}P_{3} = 8(8 - 1)(8 - 2)
_{8}P_{3} = 8(7)(6) = 336
There are 336 possible arrangements of first, second, and third places.
Notice that once there are 3 factors, you stop!
Word problem #2
Tires on your cars should be rotated at regular intervals. How many ways can four tires be arranged?
Solution
Since all four tires are being rotated, you are using all the tires.
Evaluate _{n}P_{r} with n = 4 and r = 4
_{4}P_{4} = 4(4 - 1)(4 - 2)(4 - 3)
_{4}P_{4} = 4(3)(2)(1) = = (12)(2) = 24
The number of ways to arrange four tires on a car is 24.
Word problem #3
A baseball coach is going to pick 8 players from a baseball squad of 16 to take turn batting against the pitcher. How many batting orders are possible?
Solution
The total number of batting orders is the number of ways to arrange 8 players in order from a squad of 16.
Evaluate _{n}P_{r} with n = 16 and r = 8
_{16}P_{8} = 16(16 - 1)(16 - 2)(16 - 3)(16 - 4)(16 - 5)(16 - 6)(16 - 7)
_{16}P_{8} = 16(15)(14)(13)(12)(11)(10)(9)
_{16}P_{8} = 518,918,400
Next time a baseball coach says that he had looked at all possible batting orders and picked the best ones, just say, "sure."
These permutation word problems will also show you how to use the multiplication principle to solve more complicated problems.
Word problem #4
A photographer is trying to take a picture of two men, three women, and four children. If the men, the women, and the children are always together, how many ways can the photographer arrange them?
Solution
Since the men, the women, and the children will stay together, we will have three groups. Below, see a picture of this situation.
Then, the problem has the following four tasks:
Task 1: Find the number of ways the 2 men can be arranged (_{2}P_{2})
Task 2: Find the number of ways the 3 women can be arranged (_{3}P_{3})
Task 3: Find the number of ways the 4 children can be arranged (_{4}P_{4})
Task 4: Find the number of ways the 3 groups can be arranged (_{3}P_{3})
Then, use the fundamental counting principle shown below to find the total number of permutations of all 4 tasks.
Fundamental counting principle
If you have n choices for a first task and m choices for a second task, you have n × m choices for both tasks.
Therefore, evaluate _{2}P_{2} , _{3}P_{3} , _{4}P_{4 }, and _{3}P_{3} and then multiply _{2}P_{2} , _{3}P_{3} , _{4}P_{4} , and _{3}P_{3} together.
_{2}P_{2} = 2(2 - 1) = 2(1) = 2
_{3}P_{3} = 3(3 - 1)(3 - 2) = 3(2)(1) = 6
_{4}P_{4} = 4(4 - 1)(4 - 2)(4 - 3) = 4(3)(2)(1) = 24
_{3}P_{3} = 6
_{2}P_{2} × _{3}P_{3} × _{4}P_{4} × _{3}P_{3} = 2 × 6 × 24 × 6 = 1728
The photographer has 1728 ways to arrange these people. He better not make a big fuss about it!
Word problem #5
6 boys and 8 girls will have a presentation in class today. If the teacher is going to allow the girls to go first, how many different arrangement are there for the presentation?
Solution
If the girls present first, then the number of arrangement is _{8}P_{8}
Then, when the boys present, the number of arrangements is _{6}P_{6}
Using the multiplication principle, the total number of arrangement is _{8}P_{8} × _{6}P_{6}
_{8}P_{8} ×_{ 6}P_{6} = (8 × 7 × 6 × 5 × 4 × 3 × 2 × 1)(6 × 5 × 4 × 3 × 2 × 1)
_{8}P_{8} × _{6}P_{6} = (40320)(720) = 29,030,400
Word problem #6
How many four- letter passwords can be made using the six letters a, b, c, d, e, and f?
With no repetitions, you can use the formula _{n}P_{r} = n(n - 1)(n - 2)(n - 3) ... and evaluate _{6}P_{4}.
_{6}P_{4} = 6(6 - 1)(6 - 2)(6 - 3) = 6 × 5 × 4 × 3 = 360
Notice that there are 6 choices for the first letter, 5 choices for the second letter, 4 choices for the third letter, and 3 choices for the fourth letter.
However, with repetitions, notice that there are 6 choices for the first letter, 6 choices for the second letter, 6 choices for the third letter, and 6 choices for the fourth letter.
_{6}P_{4} with repetitions = 6 × 6 × 6 × 6 = 1296
Word problem #7
How many nine-letter passwords can be made using four a's, two b's, and three c's,?
This problem requires a special formula.
Let n be the number of items to be arranged.
Let n_{1} be items that are of one kind and are indistinguishable.
Let n_{2} be items that are of another kind and are indistinguishable.
Let n_{k} be items that are of a kth kind and are indistinguishable.
Then you can use the formula you see below to find the number of distinguishable permutations:
In our problem above, there are nine letters to be arranged. So, let n = 9
Four a's are indistinguishable. So let n_{1} = 4
Two b's are indistinguishable. So let n_{2} = 2
Three c's are indistinguishable. So let n_{3} = 3
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