Permutations vs. combinations (what this calculator does)

Permutations and combinations are two closely related counting tools used in probability, statistics, and everyday “how many ways?” questions. They answer different questions:

• Permutation (nPr): counts ordered selections (order matters). Example: awarding 1st/2nd/3rd place, creating a PIN where each position is distinct, arranging speakers.
• Combination (nCr): counts unordered selections (order doesn’t matter). Example: choosing a committee, selecting lottery numbers, picking toppings where order is irrelevant.

This calculator computes nPr, nCr, or both from your inputs n (total distinct items) and r (items selected).

Key definitions

Total items (n) and selected items (r)

n is the number of distinct items available. r is how many you pick from them. Typical counting problems use integers with 0 ≤ r ≤ n.

Factorial (n!)

The factorial of a non‑negative integer n is the product of all integers from 1 to n:

n! = n × (n − 1) × … × 2 × 1, and by definition 0! = 1.

Factorials grow very fast (e.g., 20! is already about 2.43×10¹⁸), which is why results can become extremely large even for moderate inputs.

Formulas used

These are the standard closed‑form formulas for permutations and combinations (without repetition):

Permutation (order matters): nPr

The number of ordered ways to pick r items from n distinct items is:

Intuition: you have n choices for the first position, n−1 for the second, and so on, for r positions.

Combination (order doesn’t matter): nCr

The number of unordered ways to pick r items from n distinct items is:

Because each set of r chosen items can be arranged in r! ways, combinations relate to permutations via:

C(n, r) = P(n, r) / r!

Interpreting your results

• If nPr is large, it means there are many distinct ordered outcomes (rankings, sequences, arrangements).
• If nCr is large, it means there are many distinct groups you can form (teams, sets, selections) regardless of order.
• If you choose Both, you can compare how much “order” increases the count. Often nPr = nCr × r!, so order multiplies the number of outcomes by r!.

Worked example

Problem: You have n = 10 candidates and want to choose r = 3 people.

• Combination (committee of 3): order does not matter.
  C(10, 3) = 10! / (3! · 7!) = (10·9·8) / (3·2·1) = 120
• Permutation (gold/silver/bronze): order matters.
  P(10, 3) = 10! / 7! = 10·9·8 = 720

Interpretation: there are 120 possible groups of 3, but 720 possible podium outcomes because each group of 3 can be arranged into ranks in 3! = 6 ways.

Common scenarios (quick comparison)

Assumptions and limitations (important)

• Integers expected: n and r should be whole numbers. If you enter decimals, results may be invalid or rounded depending on implementation.
• Typical domain: this page is for counting without repetition and usually assumes 0 ≤ r ≤ n.
• If r > n: for “without repetition” problems, the count is 0 (you can’t pick more distinct items than you have).
• Very large values: factorial-based results become enormous quickly. Large n (even a few hundred) can exceed standard number sizes; the page may display scientific notation, big integers, or overflow depending on how results are implemented.
• Repetition allowed? Not in nPr/nCr as defined here. If your problem allows repeats (e.g., PIN digits can repeat), you need different formulas (such as n^r for ordered selections with repetition).

Helpful properties (sanity checks)

• Symmetry: C(n, r) = C(n, n − r). Choosing r items is the same as excluding n−r items.
• Edge cases: C(n, 0) = 1 and P(n, 0) = 1 (there is exactly one way to choose “nothing”).