Video: Finding the Number of Ways to Choose ๐‘› out of ๐‘š Things

In a class, there are 18 boys and 7 girls. Determine the number of ways that a team of 3 members can be selected.

02:52

Video Transcript

In a class, there are 18 boys and seven girls. Determine the number of ways that a team of three members can be selected.

In this question, weโ€™re looking to choose a team of three people from our class. Now in our class, there are 18 boys and seven girls. Weโ€™re not told that our team should be made up of a certain number of boys and girls, though. So instead, we add 18 and seven. And we see that weโ€™re looking to find the number of ways of choosing three students from a total of 25. And in fact, weโ€™re not given any indication at all that order matters here. In mathematics, this has a special name. Itโ€™s called a combination. A combination is a way of calculating the total outcomes of an event where the order of the outcomes doesnโ€™t matter.

We say that the number of ways of choosing ๐‘Ÿ items from a total of ๐‘› items, where the order of these items does not matter, is ๐‘› choose ๐‘Ÿ. ๐‘› choose ๐‘Ÿ itself is ๐‘› factorial over ๐‘Ÿ factorial times ๐‘› minus ๐‘Ÿ factorial. Now in this case, weโ€™re looking to choose three students, a team of three students, from a total of 25. And so, weโ€™re going to let ๐‘› be equal to 25 and ๐‘Ÿ be equal to three. And then, we see that the number of ways of choosing this team of three members is 25 choose three. And thatโ€™s equal to 25 factorial over three factorial times 25 minus three factorial.

Now, in fact, 25 minus three is 22. So, we simplify this a little to give us 25 factorial over three factorial times 22 factorial. Now, we know that 25 factorial is 25 times 24 times 23 and so on. But generally, we want to avoid evaluating the factorials in our formula. Instead, we notice that 25 factorial can be written as 25 times 24 times 23 times 22 factorial. And then, we see we can simplify our fraction by dividing both the numerator and denominator by 22 factorial. Similarly, we know three times two is six, and we can divide both 24 and six by six. And so, 25 choose three simplifies to 25 times four times 23 divided by one, which is, of course, simply 25 times four times 23. 25 times four is 100. And so, this becomes 100 times 23, which is 2300.

And so, if there are 18 boys and 17 [seven] girls in a class, the number of ways that a team of three members can be selected is 2300.

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