Permutations vs. combinations
Both count ways to pick r items from a set of n — the difference is whether order matters. Permutations count ordered arrangements; combinations count unordered selections.
nPr = n! ÷ (n − r)!
nCr = n! ÷ (r! × (n − r)!)
With n = 5 and r = 2: nPr = 120 ÷ 6 = 20 ordered arrangements, and nCr = 120 ÷ (2 × 6) = 10 unordered selections. There are exactly half as many combinations because each pair can be ordered two ways.
When to use which
- Permutation: order matters — race finishes, passwords, ranked picks, assigning distinct roles.
- Combination: order doesn't matter — lottery numbers, committees, hands of cards, choosing toppings.
Factorials
A factorial n! multiplies every whole number from 1 to n, and 0! = 1 by definition. Factorials grow extremely fast, so very large n produces huge results shown in scientific notation.
Frequently asked questions
What is the difference between a permutation and a combination?
Order matters in a permutation but not in a combination. Choosing a president and vice-president from a group is a permutation, because the two roles are different. Choosing a two-person committee is a combination, because the pair is the same regardless of order.
How do you calculate permutations (nPr)?
The number of permutations of r items from n is nPr = n! ÷ (n − r)!. For 5 items taken 2 at a time, that is 5! ÷ 3! = 120 ÷ 6 = 20 ordered arrangements.
How do you calculate combinations (nCr)?
Combinations divide out the duplicate orderings: nCr = n! ÷ (r! × (n − r)!). For 5 items taken 2 at a time, that is 120 ÷ (2 × 6) = 10 unordered selections — half the 20 permutations, since each pair can be arranged two ways.
What is a factorial?
A factorial, written n!, is the product of all whole numbers from 1 up to n. So 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition 0! = 1. Factorials count the number of ways to arrange n distinct items in order.
Why is nCr always less than or equal to nPr?
Every combination corresponds to r! different permutations — the ways its chosen items can be ordered. Because combinations ignore order, you divide nPr by r!, so nCr is never larger than nPr and equals it only when r is 0 or 1.
Disclaimer: Very large values are shown in scientific notation and may lose precision beyond about 15 significant digits. Provided for educational purposes.