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

Determine the number of ways to choose 3 people from 55.

02:23

Video Transcript

Determine the number of ways to choose three people from 55.

In this question, weโ€™re looking to select three people from a larger group of 55. And itโ€™s important to realize that it doesnโ€™t matter the order in which we choose these people. In mathematics, we call that a combination. And we say that to calculate the number of combinations or the number of ways of choosing ๐‘Ÿ items from a total of ๐‘› items, where order doesnโ€™t matter, we use ๐‘› choose ๐‘Ÿ. Where ๐‘› choose ๐‘Ÿ is calculated by dividing ๐‘› factorial by the product of ๐‘Ÿ factorial and ๐‘› minus ๐‘Ÿ factorial. And, of course, ๐‘› factorial is ๐‘› times ๐‘› minus one times ๐‘› minus two and so on.

Now, in this question, weโ€™re looking to choose three people from a total of 55, and so thatโ€™s 55 choose three. And whilst we could type this in on our calculator, letโ€™s look at how to use the formula. We see that ๐‘› is 55 in our example and ๐‘Ÿ is three. So we get 55 factorial over three factorial times 55 minus three factorial, which simplifies to 55 factorial over three factorial times 52 factorial.

Now, generally, we want to try and avoid fully evaluating our factorials. We know that 55 factorial is 55 times 54 times 53 times 52 and so on. So, instead, weโ€™re going to rewrite 55 factorial as the product of 55, 54, 53, and 52 factorial. And then we see that we can divide both the numerator and denominator of our fraction by 52 factorial. Letโ€™s look for some other common factors.

Well, three multiplied by two is six. And we know 54 divided by six is nine. So 55 choose three simplifies to 55 times nine times 53 all divided by one, which is simply 55 times nine times 53. 55 times nine times 53 is 26,235. And so, in evaluating 55 choose three, weโ€™ve determined the number of ways that we will choose three people from 55. Itโ€™s 26,235.

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