What each calculation counts
Permutations and combinations are two fundamental counting techniques in probability and statistics. A permutation counts the number of ways to arrange or order a selection from a larger set. A combination counts the number of ways to choose a subset without regard to order. If you're picking a 3-person committee from 10 employees, the combination tells you how many distinct committees exist. If you're assigning those 3 people to President, Vice President, and Treasurer, the permutation tells you how many ways to fill those ranked roles.
The formula
nPr = n! / (n − r)! and nCr = n! / (r! × (n − r)!)
In both formulas, n is the total number of items available, and r is the number of items you're selecting. The factorial (!) multiplies all positive integers from 1 up to that number. Notice that nCr divides by an extra r!, which removes the ordering—that's why combinations are always smaller.
Worked example
Suppose you have 8 runners in a race and want to award gold, silver, and bronze medals (positions matter). Here, n = 8 and r = 3.
For permutations (nPr):
- 8P3 = 8! / (8 − 3)!
- = 8! / 5!
- = (8 × 7 × 6 × 5!) / 5!
- = 8 × 7 × 6
- = 336 ways
There are 336 different possible medal rankings.
Now imagine instead you're simply selecting 3 runners to form a relay team (order doesn't matter). Here, n = 8 and r = 3.
For combinations (nCr):
- 8C3 = 8! / (3! × 5!)
- = (8 × 7 × 6) / (3 × 2 × 1)
- = 336 / 6
- = 56 ways
There are 56 different possible relay teams. Notice it's exactly 1/6th of the permutation result, because there are 3! = 6 ways to arrange any group of 3 people.
Common mistakes to avoid
Confusing the scenarios: The biggest trap is using permutations when combinations are needed, or vice versa. Always ask: does the order or arrangement matter? If yes, use nPr. If no, use nCr.
Forgetting that r ≤ n: You cannot select 5 items from a set of 3. If your inputs violate this rule, the calculator will flag an error.
Mishandling zero: 0! = 1 by definition. This is important for edge cases like nC0 (choosing nothing from n items) or nPn (arranging all n items), both of which equal 1.
Mixing up the formula: A common slip is writing nCr with only one factorial in the denominator, or forgetting the (n − r)! term in nPr. Double-check that your denominator matches the formula above.