Key takeaways
- nCr = n! ÷ (r! × (n − r)!) counts groups where order does not matter.
- Shortcut: multiply r numbers counting down from n, then divide by r!.
- nCr = nC(n − r). Choosing 3 to keep is the same as choosing 7 to leave out.
- If order matters, use permutations (nPr) instead.
In this guide
What is a combination?
You have 5 friends and can invite 2 to the cinema. Inviting Asha and Ben is the same as inviting Ben and Asha. Because the order does not matter, this is a combination.
The notation nCr, read 'n choose r', means the number of ways to choose r items from n.
Why do combinations matter?
Counting by hand works for small groups, but the numbers explode quickly. Combinations let you count millions of possible groups in a few seconds, which is the first step in working out odds and probabilities.
There are 2,598,960 different five-card poker hands from a 52-card deck. You could never list them, but nCr counts them in one line.
The formula
| Symbol | What it means | Example |
|---|---|---|
| n | Total items to choose from | 10 toppings |
| r | How many you choose (0 to n) | 3 toppings |
| r! | Removes the orderings of each group | 3! = 6 |
| (n − r)! | Cancels the items you did not pick | 7! |
| nCr | Number of different groups | 120 |
The r! on the bottom is the key idea. Every group of 3 can be ordered in 3! = 6 ways, but we only want to count it once.
How to calculate combinations step by step
- 1Write down n and r. Check that r is between 0 and n.
- 2Count down r numbers from n and multiply them. For 10C3:
10 × 9 × 8 = 720. - 3Work out r! For r = 3:
3 × 2 × 1 = 6. - 4Divide.
720 ÷ 6 = 120. That is nCr.
Now enter your own n and r in the calculator below and see the full working.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Choosing r bigger than n. You cannot choose 7 items from 5. The calculator shows an error.
- Forgetting to divide by r!. Without it you count each group many times.
- Working out huge factorials in full. 49! is enormous. Cancel first with the shortcut.
Tips and tricks
| Order matters? | Example | Formula | |
|---|---|---|---|
| Combination | No | Team of 3 from 10 | nCr = 120 |
| Permutation | Yes | Gold, silver, bronze from 10 | nPr = 720 |
- Link to permutations:
nPr = nCr × r!. So 10P3 =120 × 6 = 720. - nC0 and nCn are always 1. There is one way to pick nothing and one way to pick everything.
- nC1 is always n. Picking one item from n can be done n ways.
Where you'll use it in real life
- Lotteries: 49C6 = 13,983,816 possible tickets.
- Card games: 52C5 = 2,598,960 possible poker hands.
- Teams and committees: choosing 4 players from a squad of 12.
- Probability: the nCk part of the binomial probability formula.
Quick summary
Combinations count the groups you can choose when order does not matter: nCr = n! ÷ (r! × (n − r)!). The fastest way is to multiply r numbers down from n and divide by r!. Use the calculator above to find any nCr with every step shown.
Worked example
A pizza shop offers 10 toppings and a special deal lets you pick any 3. The order you name them does not change the pizza. How many different 3-topping pizzas are possible?
- 01nCr = n! / (r! × (n - r)!)
- 0210C3 = 10! / (3! × 7!)
- 0310C3 = 120
These numbers are pre-filled in the calculator above.
Frequently asked questions
What is the difference between nCr and nPr?
nCr counts groups where order does not matter. nPr counts arrangements where order matters. nPr = nCr × r!, so nPr is always at least as big.
What is nC0 or nCn?
Both equal 1. There is one way to choose nothing and one way to choose everything.
How many ways can you choose 6 numbers from 49?
49C6 = 13,983,816. That is why the chance of winning such a lottery with one ticket is about 1 in 14 million.
How many handshakes happen if 6 people all shake hands once?
Each handshake is a pair, and order does not matter, so it is 6C2 = (6 × 5) ÷ 2 = 15 handshakes.
