permutation combination calculator

What is a permutation vs. combination?

A permutation counts selections where order matters. A combination counts selections where order does not matter.

If you are assigning gold, silver, and bronze medals to 3 people out of 10 finalists, order matters, so use permutation. If you are simply choosing a 3-person committee out of 10 people, order does not matter, so use combination.

How to use this permutation combination calculator

• Enter the total number of items as n.
• Enter how many items are being selected as r.
• Click Calculate to get both nPr and nCr instantly.
• Use Clear to reset and run another scenario.

Formulas explained

Permutation formula (nPr)

nPr = n! / (n - r)! This removes all arrangements that are not part of the selected group size r.

Combination formula (nCr)

nCr = n! / (r!(n - r)!) Since combinations ignore order, we divide out the r! duplicate arrangements of each group.

Example calculation

Suppose n = 10 and r = 3:

• 10P3 = 10 × 9 × 8 = 720
• 10C3 = (10 × 9 × 8) / (3 × 2 × 1) = 120

This means there are 720 ordered outcomes, but only 120 unique unordered groups.

Common mistakes to avoid

• Using permutation when order does not matter.
• Using combination when order does matter.
• Entering r greater than n (not valid for standard nPr/nCr).
• Forgetting that 0! = 1.

Where these calculations are used

Permutations and combinations are used in probability, statistics, machine learning, cryptography, exam planning, scheduling, and game strategy. Anytime you need to count outcomes without listing every case manually, nPr and nCr help you move faster and more accurately.

Quick takeaway

Remember this rule of thumb: Order matters = permutation. Order does not matter = combination.

Keep this page bookmarked as your go-to nPr nCr calculator for fast, reliable counting.