How to Use nCr and nPr (Combinations vs Permutations)
nCr and nPr both count ways to pick r items from a group of n. The difference is whether the order of selection matters. nCr ignores order — choosing apples then oranges is the same as oranges then apples. nPr cares about order — first place and second place in a race are different.
The two formulas
nCr(n, r) = n! / (r! × (n − r)!). nPr(n, r) = n! / (n − r)!. The only difference is the extra r! in the denominator of nCr — it removes the orderings of the r chosen items. Multiply nCr by r! to get nPr: nPr = nCr × r!. So nPr is always greater than or equal to nCr.
Worked example — combinations
How many ways to pick 3 students from a class of 10 for a project? Order does not matter (the project does not care who was picked first). nCr(10, 3) = 10! / (3! × 7!) = 720,720 / (6 × 5,040) = 120. There are 120 different teams of three.
Worked example — permutations
How many ways to award gold, silver, bronze medals from 10 finalists? Order matters (gold is different from silver). nPr(10, 3) = 10! / 7! = 10 × 9 × 8 = 720. There are 720 different podium arrangements — exactly 6 (= 3!) times the number of teams from the previous example.
Calculator method
Type the n value, press nCr (or nPr), type the r value, press equals. Some calculators show it as 10 nCr 3 in the expression line. Online calculators usually take comma-separated arguments: nCr(10, 3). The button often sits on the SHIFT layer because it is less common than the basic functions.
Edge cases
nCr(n, 0) = 1 — there is one way to pick nothing (the empty selection). nCr(n, n) = 1 — there is one way to pick everything. nCr(n, 1) = n — n single items to pick. nCr(n, r) = nCr(n, n−r) — picking 3 out of 10 to include is the same as picking 7 out of 10 to exclude. The symmetry shortens many calculations.
Common use cases
Lottery odds — nCr(49, 6) = 13,983,816 different 6-number tickets in a 49-ball draw. Poker hands — nCr(52, 5) = 2,598,960 five-card hands from a deck. Class committees, team selections, taste-test panels, and any 'how many ways can I choose' question. nPr handles ordered cases: race podiums, lock combinations (the word combination is misleading — a lock combination is technically a permutation), seating arrangements, and password counting.
Frequently asked questions
What is the difference between combinations and permutations?
Combinations count selections where order does not matter — picking 3 friends for a dinner. Permutations count arrangements where order does matter — first, second, and third place in a race. nCr counts combinations; nPr counts permutations. Both use factorials; nPr is always greater than or equal to nCr.
When should I use nCr vs nPr?
Ask: 'Does the order I pick the items matter for the final outcome?' If yes (race positions, seat assignments, ranked lists), use nPr. If no (committees, hands of cards, project teams), use nCr. The wording of the problem usually signals which one — phrases like 'arrangement', 'ranked', or 'in order' point to nPr.
What is 5 choose 2?
5 choose 2 — written nCr(5, 2) or C(5, 2) or ⁵C₂ — equals 10. There are 10 unordered pairs you can make from five items. The formula: 5! / (2! × 3!) = 120 / 12 = 10. Press 5, nCr, 2, equals on a calculator.
How do I enter nCr on an online calculator?
Click the nCr button (sometimes labelled C(n,r) or ⁿCᵣ). It usually opens a bracket: nCr(. Type the n value, comma, the r value, close bracket, press equals. Some interfaces ask for n and r separately in two prompts. The result is always a positive integer.
Why is nCr(n, 0) equal to 1?
There is exactly one way to choose nothing — pick no items. The empty selection counts as a valid choice. Mathematically, the formula gives n! / (0! × n!) = 1 (because 0! is defined as 1). The convention keeps formulas tidy at edge cases.