Permutation and Combination Calculator

What is a permutation and combination calculator?

A permutation and combination calculator tells you how many ways you can select r items from a set of n items — once when order matters and once when it does not. Enter two whole numbers, n (the total) and r (how many you pick), and the tool returns both results at once: the number of permutations (nPr) and the number of combinations (nCr).

The key distinction is order. A permutation counts ordered selections, so arranging the same items differently produces a new outcome. A combination counts unordered selections, so only the membership of the chosen group matters. Picking a president, vice-president and treasurer is a permutation; picking any three committee members is a combination.

How does the calculator work?

The calculator applies two related formulas:

nPr = n! / (n−r)!

nCr = n! / (r!·(n−r)!)

where the terms mean:

• n — the total number of distinct items available to choose from.
• r — how many items you are selecting; it must be a whole number between 0 and n.
• n! — n factorial, the product n × (n−1) × (n−2) × … × 1.
• r! — r factorial, used to divide out the orderings within the chosen group.

Rather than computing huge factorials and then dividing, the tool evaluates each result as a running product, multiplying only the terms that survive the cancellation. For nPr it multiplies the top r descending terms — n × (n−1) × … × (n−r+1) — so moderate values stay exact and do not overflow. nCr is then found by dividing that permutation count by r!.

Both results are pure counts, so they are unitless whole numbers. Because every combination of r items can be reordered r! ways, the two are linked by nPr = nCr × r! — permutations are always equal to or larger than combinations for the same inputs.

Examples

Each example below matches the calculator exactly. Type the inputs into the tool above to confirm.

Example 1: choosing 3 from 10

With n = 10 and r = 3, the permutation count multiplies the top three descending terms:

nPr = 10 × 9 × 8 = 720

For combinations, divide by 3! to remove the orderings:

nCr = 720 ÷ 3! = 720 ÷ 6 = 120

So there are 720 ordered ways and 120 unordered ways to pick 3 items from 10.

Example 2: selecting all items (r = n)

With n = 5 and r = 5, every item is chosen:

nPr = 5 × 4 × 3 × 2 × 1 = 120

nCr = 1

There are 120 ways to arrange all five items, but only 1 way to choose the whole set — once you take everything, there is nothing left to vary.

Example 3: a small pair

With n = 6 and r = 2:

nPr = 6 × 5 = 30

nCr = 30 ÷ 2! = 30 ÷ 2 = 15

That is 30 ordered pairs and 15 unordered pairs from six items — exactly half, because each pair can be ordered two ways.

nPr and nCr reference table

The table shows both counts for selecting r items from a fixed set of n = 6, so you can see how permutations and combinations diverge as r grows.

n	r	nPr (ordered)	nCr (unordered)
6	0	1	1
6	1	6	6
6	2	30	15
6	3	120	20
6	4	360	15
6	5	720	6
6	6	720	1

Notice the symmetry in the nCr column: choosing 2 of 6 (15 ways) matches choosing 4 of 6 (15 ways), because leaving 4 behind is the same as picking 2. The nPr column never decreases as r rises, since each new pick adds more orderings.

Common uses

• Lottery and game odds. Combinations count how many ticket choices exist (order does not matter), the denominator behind “how many combinations” questions.
• Committees and teams. Use nCr to count groups where roles are interchangeable, and nPr when seats or titles are distinct.
• Passwords and PINs. Permutations count ordered codes, where 1-2-3 differs from 3-2-1.
• Seating and scheduling. Arranging people in a row or events in a timetable is a permutation problem.
• Probability setup. Many probability problems reduce to a count of favourable outcomes over total outcomes, each found with nPr or nCr.

Tips and common mistakes

• Ask “does order matter?” first. If rearranging the same items counts as a new outcome, use permutations (nPr); if not, use combinations (nCr).
• Keep r within range. r must be a whole number from 0 to n. Values where r is greater than n are impossible and not allowed.
• Remember the edge cases. When r = 0 or r = 1, nPr and nCr are equal because r! is 1. When r = n, nCr is always 1.
• Use the bridge formula to check. If nPr ÷ r! does not equal your nCr, one of the figures is off — the two must satisfy nPr = nCr × r!.
• Do not confuse nCr with raw factorials. nCr divides out both the unchosen items and the internal ordering, which is why it stays far smaller than n!.

Limitations and notes

The calculator handles items without repetition and treats every item as distinct — each can be chosen at most once, and no two are identical. Problems that allow repeats (such as rolling the same die value twice) or that contain duplicate items use different formulas and are not covered here.

Because results are built as a running product, moderate inputs stay exact, but extremely large values of n and r grow astronomically and may exceed what standard number handling can represent precisely. For everyday counting, scheduling and odds questions the figures are exact. This tool is intended for educational and general analytical use.

Related tools

For the spread and centre of a data set, use the mean, median & mode calculator and the standard deviation calculator, and browse more in the statistics category.