Statistics How To

5 Choose 3 or 5C3: Figuring out Combinations

Probability and Statistics > Probability Index > 5 Choose 3

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.
combinations
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!

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5 Choose 3 or 5C3: Figuring out Combinations was last modified: October 23rd, 2017 by Stephanie Glen

6 thoughts on “5 Choose 3 or 5C3: Figuring out Combinations

  1. Richard Ward

    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.

  2. Andale Post author

    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

  3. Andale Post author

    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.