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
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.