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. Now we should see quite quickly
that the order here doesn’t matter. For example, let’s say three of her
topics are fractions, decimals, percentages. She could choose them in that
order: fractions, decimals, percentages. She could alternatively say
fractions first and then choose percentages and then decimals. There are in fact six different
ways that she could choose these topics. But we see that choosing fractions,
decimals, percentages would be exactly the same as selecting decimals, then
percentages, then fractions. When we want to choose a number of
items from a larger group and order doesn’t matter, these are called
combinations.
Now, in order to find the number of
combinations, we’re going to begin by thinking about permutations. Now, permutations occur when order
does matter. So if we think about our earlier
example, where we ordered the three topics, there was only one combination but six
different permutations. We might recall that đť‘›Pđť‘ź is the
number of ways of choosing đť‘ź items from a selection of đť‘› when order does
matter. It’s the number of
permutations. And we calculate this by working
out 𝑛 factorial divided by 𝑛 minus 𝑟 factorial. So let’s begin by working out the
number of permutations of five topics from a total of eight. That’s eight P five. That’s eight factorial over eight
minus three factorial, which is 6720. There are 6720 permutations.
