Video: Finding the Number of Ways to Choose 𝑛 out of 𝑚 Things

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 𝑛P𝑟 is the number of ways of choosing 𝑟 items from a selection of 𝑛 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.