Video Transcript
Olivia was asked by her teacher to
choose five from the eight topics given to her. How many different five-topic
groups could she choose?
In this question, we’re looking at
how many ways we can choose five items from a group of eight. And here order doesn’t matter. For example, let’s say that three
of the topics she can choose from are fractions, decimals, and percentages. She could choose fractions first,
then decimals, then percentages. She could alternatively still
choose fractions first but then choose percentages and then decimals. If we list all of these different
ways out, there are six different ways in which she could choose the three
topics. But of course within this context,
choosing fractions, decimals, and percentages is actually the same as choosing them
in any other order. In fact, when we want to choose a
number of items from a larger group, we call this a combination.
To find a formula, we’ll begin then
by thinking about permutations. This is when the order does
matter. In this case, there are six
permutations and just one combination. Now let’s say 𝑛𝑃𝑟 is the number
of ways of choosing 𝑟 items from a selection of 𝑛 when order does matter. We might recall that the formula we
use to calculate 𝑛𝑃𝑟 is 𝑛 factorial over 𝑛 minus 𝑟 factorial. In this case, we’re interested in
the number of ways of choosing five topics from a total of eight. So we’re going to calculate eight
𝑃 five. By letting 𝑛 be equal to eight and
𝑟 be equal to five, the number of permutations here is eight factorial over eight
minus five factorial. That’s eight factorial over three
factorial, which is 6720.
So, if order does matter, there
would be 6720 ways to choose five from the eight topics given. But we know the order doesn’t
matter here. And so we need to find a way to get
rid of the extra permutations. Let’s go back to our earlier
example of choosing just three topics. To choose three topics from a total
of six when order matters, we calculate six 𝑃 three, which gives us the six
permutations we’re expecting.
To get rid of the extra
permutations, we have to divide by three factorial since we’re looking at three
subjects. That’s six divided by three
factorial, or six divided by six, which is equal to one. That gives us the one combination
we know we were expecting. In this case, since there are five
topics, we’re going to divide eight 𝑃 five by five factorial. And that gives us an answer of
56. Olivia can choose 56 different
five-topic groups from the total of eight.
