Permutation or combination? Ask whether swapping names changes the outcome
Order is the deciding distinction. The same people and the same selection size can produce very different counts when roles are involved.
Permutation and combination calculator →A committee and two officers differ
From five distinct people, choosing a two-person committee gives C(5,2)=10 possibilities. Assigning a chair and a secretary gives P(5,2)=20 possibilities because Alice-chair/Bob-secretary differs from Bob-chair/Alice-secretary. Enter n=5 and r=2 to see both counts. Ask what one complete outcome looks like before deciding which displayed result applies.
Why the factor is r factorial
Every group of r distinct objects can be ordered in r! ways. Therefore P(n,r)=C(n,r)×r!. With r=2 the factor is 2; with r=3 it is 6. The combination formula removes those duplicate orderings. This is why dividing an ordered count by r, rather than by r!, fails once r is larger than 2.
Check the tool’s assumptions
This calculator selects without replacement from distinct objects and requires integer inputs 0≤r≤n≤1000. It does not automatically handle repeated letters, identical objects or rules excluding certain pairs. Choosing zero objects has one outcome, the empty selection, rather than zero outcomes. If your situation has extra constraints, build those into the counting model before treating the basic displayed count as the answer. Listing a tiny case by hand can clarify the interpretation: from A, B and C, two-person groups are AB, AC and BC, but ordered pairs also include BA, CA and CB.
Try the calculators
- Permutation and combination calculator
Count ordered and unordered selections of r objects from n distinct objects.
- Random number
Generate a random integer in an inclusive range.
- Scientific calculator
Arithmetic, powers, trigonometry and logarithms.