Probability > 4 Choose 2 ; 4 Choose 3

Combinations are a way to figure out how many possible ways you can order a set of numbers that you pick from a larger set of numbers. The formula is:

The “!” sign is a factorial. In order to solve a factorial, just multiply the number by every number below it until you get to 1. For example, 3 factorial is 3 x 2 x 1 = 6.

**Sample combinations questions:
**

## 4 Choose 2

**Question**: How many different combinations do you get if you have 4 items and choose 2?

**Answer: **Insert the given numbers into the combinations equation and solve. “n” is the number of items that are in the set (4 in this example); “r” is the number of items you’re choosing (2 in this example):

C(n,r) = n! / r! (n – r)!

= 4! / 2! (4 – 2)!

= 4! /2! * 2!

= 4 x 3 x 2 x 1 / 2 x 1 * 2 x 1

= 24 / 4

= 6

The solution is 6. Here’s the full list of possible combinations:

{1,2}

{1,3}

{1,4}

{2,3}

{2,4}

{3,3}

{3,4}

**Note: {1,1}, {2,2}, {3,3} and {4,4} aren’t included in the list, because with combinations you can’t choose the same item twice for the same set.
**

## 4 Choose 3

How many different combinations do you get if you have 4 items and choose 3?

**Answer: **Insert the given numbers into the combinations equation and solve. “n” is the number of items that are in the set (4 in this example); “r” is the number of items you’re choosing (3 in this example):

C(n,r) = n! / r! (n – r)! =

= 4! / 3! (4 – 3)!

= 4 x 3 x 2 x 1 / 3 x 2 x 1 x 1

= 24 / 6

= 4

The solution is 4. The possible combinations are:

{1,2,3}

{1,2,4}

{1,3,4}

{2,3,4}

**Note**: Sets like {1,1,2) or {3,3,3} aren’t included in the calculation, as you can’t choose an item more than once for a set.

## 4 Choose 0

4 Choose 0 is 1.

Why? This might seem like a mind bender; how can you choose none and still get 1? But you have to look at it a slightly different way. If you have 4 items and you’re not choosing any, you still have those four items {1,2,3,4). In other words, you didn’t take any away, so you still have them all, i.e. 1 set.

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The 4 choose 0 part is confusing. So we are dividing the sets we have? So 4 choose 3 will also have this main set(the one that was given)?

and does it also mean 4 choose 0 and 4 choose 4 mean the same thing?

4 choose 4 from what you say should be {1, 2, 3, 4}. As you also mentioned 4 choose 0 has the untouched main set {1, 2, 3, 4}. So does that mean 4 C 0 === 4 C 4?

There’s no division going on…

4 choose 4 is not the same as 4 choose 0. would be all the ways to choose 4 items. i.e. 1234, 1243, 1324…etc.