Counting Theory Lesson

In this lesson, we want to learn how to solve word problems that involve permutations or combinations. Let's begin by listing the formula for each scenario:

Permutations of n Elements Taken r at a time

$$P(n, r)=\frac{n!}{(n - r)!}$$

Combinations of n Elements Taken r at a Time

$$C(n, r)=\frac{n!}{(n - r)!r!}$$

Solving Word Problems with Combinations and Permutations

Now that we have listed our formulas, let's discuss the key to this section. When we want to know the number of ways of selecting r items out of n items and repetitions are not allowed, meaning once something is selected, it can't be selected again, we will use the permutations formula when the order is important, and the combinations formula when the order is not important. Let's look at a few examples.
Example #1: Solve each word problem.
A group of 30 students are going to compete in a swimming competition. The three fastest students will earn gold, silver, and bronze medals. How many different ways can the medals be awarded to the students?
For this problem, ask the simple question: does the order matter to the end result? Let's think about this, suppose John, Jacob, and Sarah finished first, second, and third. This means John gets the gold, Jacob gets the silver, and Sarah gets the bronze. Suppose the same three students finished in the top three, but now the order is changed, so Sarah finishes first, John finishes second, and Jacob is last. Did this change the result? Yes, now Sarah gets the gold, John gets the silver, and Jacob gets the bronze. Since changing the order changes the result, we will use the formula for permutations. We have 3 winners out of a group of 30 students: $$P(30, 3)=\frac{30!}{27!}=24,360$$ This tells us there are 24,360 different ways that the medals can be awarded to 3 of the 30 students.
Example #2: Solve each word problem. A group of 45 students are going to compete in a half marathon. The first 6 students to finish the race will advance to the state championship race. How many different ways can 6 students advance to the state championship race?
For this problem, ask the simple question: does the order matter to the end result? Let's again think about this with a few sample names. Suppose we had the following 6 students that finished as the top 6 racers:

1. John
2. Larry
3. Beth
4. Amanda
5. Barry
6. Claire

What happens if the order changes? Does it change the result?

1. Claire
2. John
3. Amanda
4. Barry
5. Beth
6. Larry

The answer is no, even though their rank for this race will change, the result is that they advance to the state championship. This happens as long as the student finishes in the top 6. For this problem, we will use the combination formula since the order doesn't matter. We will have 6 students who advance out of a total of 45. $$C(45, 6)=\frac{45!}{39!6!}=8{,}145{,}060$$ This tells us that there are 8,145,060 different ways that 6 students of the 45 can advance to the state championship race.

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