Determine the number of ways to
choose three people from 55.
In this question, we’re looking to
select three people from a larger group of 55. And it’s important to realize that
it doesn’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 𝑟 items from a total of 𝑛
items, where order doesn’t matter, we use 𝑛 choose 𝑟. Where 𝑛 choose 𝑟 is calculated by
dividing 𝑛 factorial by the product of 𝑟 factorial and 𝑛 minus 𝑟 factorial. And, of course, 𝑛 factorial is 𝑛
times 𝑛 minus one times 𝑛 minus two and so on.
Now, in this question, we’re
looking to choose three people from a total of 55, and so that’s 55 choose
three. And whilst we could type this in on
our calculator, let’s look at how to use the formula. We see that 𝑛 is 55 in our example
and 𝑟 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, we’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. Let’s look for some other common
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, we’ve determined the number of ways that we will choose three people from
55. It’s 26,235.