Permutation and Combination Calculator

Calculate permutations (nPr) and combinations (nCr) with or without replacement, view detailed formulas, and generate all possible subset arrangements.

Permutation & Combination Solver

Enter the total set size n and sub-set size r to compute order-sensitive permutations and order-independent combinations.

Total Amount in a Set (n):

Amount in each Sub-Set (r):

Repetition / Replacement Mode:

List all actual combinations & permutations (max limit n ≤ 10)

How To Use the Permutation and Combination Calculator

Permutations and combinations belong to combinatorics, a branch of mathematics studying discrete structures:

Permutations (nPr): Used when the order of selection matters (e.g., electing a President and Vice President, lock codes).
Combinations (nCr): Used when the order does NOT matter (e.g., picking 2 team captains from 11 players, lottery tickets).
Without Replacement: Each element can be picked only once. Once chosen, it is removed from the remaining set.
With Replacement: Elements can be selected multiple times in a single subset (e.g., rolling dice repeatedly).

Frequently Asked Questions (FAQs)

The key difference is order. Permutations consider order (e.g., 1-2-3 is different from 3-2-1). Combinations ignore order (e.g., 1-2-3 and 3-2-1 are considered the exact same subset).

For a set of size n and sample size r:
• Permutation: nPr = n! / (n - r)!
• Combination: nCr = n! / [ r! × (n - r)! ]

When repetition is allowed:
• Permutation with replacement: P(n, r) = n^r
• Combination with replacement: C(n, r) = (r + n - 1)! / [ r! × (n - 1)! ]

A standard combination lock requires numbers to be entered in a specific sequence (e.g., 4-12-28). Because sequence order matters, it is mathematically a permutation lock.

Without replacement, choosing r items from n when r > n is impossible, yielding 0 arrangements. With replacement, you can choose r > n items because elements can be repeated endlessly.