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

Video Transcript

How many three-cards hands can be chosen from a deck of 52 cards?

In this question, we’re looking to select a hand of three cards from a collection of 52. It’s important to notice that the order in which we select these three cards does not matter. So that we could choose a jack, a 10, and an eight in any order.

In mathematics, we call this a combination. And like we said, we don’t care what order we choose the three-card hands in. So, we’re going to consider combinations when the order doesn’t matter. The number of ways of choosing 𝑟 items from a total of 𝑛 items is 𝑛 choose 𝑟. And whilst we can use a calculator to work out 𝑛 choose 𝑟, we should know it’s equal to 𝑛 factorial over 𝑟 factorial times 𝑛 minus 𝑟 factorial.

In this case then, we’re looking to choose three cards from a total of 52. So, we’re going to work out 52 choose three. According to our formula, that’s 52 factorial over three factorial times 52 minus three factorial. And since 52 minus three is 49, this simplifies a little bit to 52 factorial over three factorial times 49 factorial.

Now, wherever possible, if we’re trying to calculate these sort of problems, we should avoid writing out the full expansion of 52 factorial. That’s 52 times 51 times 50 and so on. Similarly, we should probably avoid writing out 49 factorial. That’s 49 times 48 times 47 and so on. Instead, we spot that 52 factorial is the same as 52 times 51 times 50 times 49 factorial. And then, we can divide through by this common factor of 49 factorial.

Let’s look for more common factors on our numerator and denominator. We can divide both three and 51 by three. And similarly, we can divide two and 50 by two. And so, we see 52 choose three simplifies to 52 times 17 times 25 all divided by one or just 52 times 17 times 25. And this equals 22,100. We’re interested in 22,100 combinations. We can choose 22,100 three-card hands from a deck of 52 cards.