Probability and Statistics > Probability Index > 5 Choose 3

5C3 or 5 choose 3 refers to how many combinations are possible from 5 items, taken 3 at a time. What is a combination? Just the number of ways you can choose items from a list. For example, if you had a box of five different kinds of fruit and you could choose 2, you might get an apple and an orange, an orange and a pear, or a pear and and orange. But how many combinations are possible?

Watch the video or read on below:

## What are Combinations?

Combinations are just the full spread of different ways you can arrange the various subsets of one larger set. For a simple example take the set of A={1,2}. You can form four subsets of this set: {}, {1}, {2}, and {1,2}. These subsets are thus the combinations of set A.

Ironically, combinations have nothing to do with combination locks! One important fact about combinations is that order doesn’t matter. So in real life, a combination lock should be called a “permutation lock”, because the order *does* matter when you’re entering numbers into a lock.

## Permutations and Combinations

You always have fewer combinations that permutations, and here’s why:

Take the numbers 1,2,3,4. If you want to know how many ways you can select 3 items where the order doesn’t matter (and the items aren’t allowed to repeat), you can pick:

123

234

134

124

However, if you want permutations (where the order *does* matter, the same set has 24 different possibilities. Just take the first combination, 1,2,3 and think of the ways you can order it.

123

132

321

312

231

213

There are six ways to order the numbers, which means there are 4 x 6 ways to order the set of four numbers.

## Combinations and Groups

Combinations don’t have to involve numbers — sets can also refer to groups:

Jane, Lin, Gina and Sally is a set of 4.

Pluto, Venus, Mars, Earth and Saturn is a set of 5.

Why do we care about combinations in real life? Combinations have hundreds (possibly, thousands) of applications, the most obvious of which is gambling:

Lottery organizations need to know how many ways numbers can be chosen in order to calculate odds.

Slot machine manufacturers need to know how many ways the pictures on the wheels can line up, to calculate odds and prize money.

## 5 Choose 3: Example

Find 5C3 from Al, Betty, Charlie, Delilah, Erin.

The number of possible ways you could take 3 people from that list (Al, Betty, Charlie, Delilah, Erin) are:

Al/Betty/Charlie, Al/Betty/Delilah, Al/Betty/Erin, Al/Charlie/Delilah, Al/Charlie/Erin, Al/Delilah/Erin, Betty/Charlie/Delilah, Betty/Charlie/Erin, Betty/Delilah/Erin, Charlie/Delilah/Erin.

So 5 choose 3 = 10 possible combinations.

However, there’s a** shortcut** to finding 5 choose 3. The combinations formula is:

nCr = n! / (n – r)! r!

n = the number of items.

r = how many items are taken at a time.

The ! symbol is a factorial, which is a number multiplied by all of the numbers before it. For example, 4! = 4 x 3 x 2 x 1 = 24 and 3! = 3 x 2 x 1 = 6.

So for 5C3, the formula becomes:

nCr = 5!/ (5 – 3)! 3!

nCr = 5!/ 2! 3!

nCr = (5 * 4 * 3 * 2 *1) / (2 * 1)(3 * 2 * 1)

nCr = 120 / (2 * 6)

nCr = 120 / 12

nCr = 10

## 5 Choose 3: The easier way

Figuring out factorials is reasonably easy for smaller digits, like 5 or 3. But what if you are trying to choose 3 people from a group of 120? 120! becomes 120 x 119 x 188 x… which is going to take you *forever* to figure out. The solution? Use a calculator. Most scientific and graphing calculators can evaluate factorials for you, but you don’t even need a hand-held calculator. You have a couple of options on the internet.

- Google will evaluate factorials. Type 5! into Google search and it will give you 120 as an answer on the Google calculator.
- Use our online combinations calculator. It not only gives you the result — it gives you the working out too!

**Note**: although the C in “5c3” is often written as “choose,” it actually means *Combination*!

If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you’re are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

Cool

BEWARE ……

The worked example in the video at the top got the wrong answer!

5!/3!(5-3)!=10, NOT 5 – the expansion shown later gives the correct answer.

Hi, Richard,

Thanks for your comment! Where do you see the answer as “5”? I checked the video and it does give the correct answer as 10.

Stephanie

I think I see it…are you talking about the screen shot? That’s a screen shot of 1/2 way through the video. I can see how that might be confusing so I changed the screen shot in YouTube. It should show up soon.

How about if we’re looking for n? Using nC3=42.

Use algebra to solve the combinations equation with the parts you know: n! 3! (n-3)! = 42