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

Question Video: Finding the Number of Ways to Choose 𝑛 out of 𝑚 Things Mathematics • Third Year of Secondary School

Olivia was asked by her teacher to choose 5 from the 8 topics given to her. How many different five-topic groups could she choose?

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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.

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