Permutation & Combination Calculator · Practical guide
How Many Ways to Choose 3 People From 10
120 possible committees, 720 possible officer slates — the difference is whether order matters.
At a glance
The Answer: 120 Ways
Choosing 3 people from a group of 10 gives 120 different groups when the order of selection does not matter — a combination, written C(10,3). The formula is nCr = n! ÷ (r! × (n − r)!), so C(10,3) = 10! ÷ (3! × 7!). The 7! cancels most of 10!, leaving (10 × 9 × 8) ÷ (3 × 2 × 1) = 720 ÷ 6 = 120.
Interactive tool
Permutation & Combination Calculator
Enter your own values, then compare the result with the examples below.
Results
Formulas:
nPr = n! / (n - r)!
nCr = n! / (r! × (n - r)!)
When Order Matters: 720 Ways
If the three chosen people get distinct roles — president, vice president, treasurer — then the order matters and you need a permutation instead: P(10,3) = 10! ÷ 7! = 10 × 9 × 8 = 720. Every unordered group of 3 can be arranged 3! = 6 ways, which is why 720 is exactly 6 times 120.
Choosing r People From 10
| r (chosen) | Combinations C(10,r) | Permutations P(10,r) |
|---|---|---|
| 1 | 10 | 10 |
| 2 | 45 | 90 |
| 3 | 120 | 720 |
| 4 | 210 | 5,040 |
| 5 | 252 | 30,240 |
Combinations peak in the middle: there are more ways to choose 5 from 10 (252) than to choose 9 from 10 (10), because choosing 9 is the same as choosing the 1 person to leave out. Enter any n and r in the calculator above to see nPr, nCr, and n! together.
Frequently Asked Questions
What if the three people get different roles?
Then order matters and you use a permutation: P(10,3) = 10 × 9 × 8 = 720 possible assignments.
Why do you divide by 6 in C(10,3)?
Because each group of 3 people can be arranged 3! = 6 ways, and those 6 orderings all represent the same committee. Dividing by 6 removes the duplicates.
Can r be larger than n?
No. You cannot choose 11 people from a group of 10, so r must be less than or equal to n, and both must be non-negative integers.
How many ways are there to choose all 10?
Exactly one: C(10,10) = 1. There is only one way to take the whole group.