What is the difference between a Permutation and a Combination?

The key difference is order. In permutations (nPr), the order of elements matters (e.g., race podium or passcode). In combinations (nCr), the order does not matter (e.g., poker hand or lottery numbers).

What are the formulas for nPr and nCr?

Permutations: nPr = n! / (n - r)!. Combinations: nCr = n! / [r! × (n - r)!].

How do you calculate combinations with repetition allowed?

When choosing r items from n types with repetition allowed, the multiset formula is: (n + r - 1)! / [r! × (n - 1)!].

Permutation and Combination Calculator - nPr, nCr & Probability

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Total Set Size (n ≥ 0):

Subset / Sample Size (r ≤ n):

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Permutations (nPr - Order Matters)

336

Combinations (nCr - Order Unimportant)

Permutations with Repetition (nʳ):	512
Combinations with Repetition:	120
Factorial n! (8!):	40,320
Factorial r! (3!):	6
Factorial (n - r)! (5!):	120

Pascal's Triangle Row (n = 8):

[1, 8, 28, 56, 70, 56, 28, 8, 1]

📝 Step-by-Step Factorial Simplification

Comprehensive Guide to Permutations, Combinations & Probability

Combinatorics is the branch of mathematics dealing with counting, arrangements, and selection configurations. Whether determining cryptographic password keyspace, genetics allele assortments, or poker probabilities, understanding the difference between permutations and combinations is essential.

Permutations (Order Matters): nPr = n! / (n − r)!
Combinations (Order Unimportant): nCr = C(n, r) = n! / [ r! × (n − r)! ]

When to Use Permutations vs. Combinations

Use Permutation (nPr): When ranking, sequence, or position matters. Examples: Locker code sequences (1-2-3 ≠ 3-2-1), 1st, 2nd, and 3rd place race finishes, or elected officer roles (President, VP, Treasurer).
Use Combination (nCr): When grouping without respect to order. Examples: Picking a 5-person committee from 20 candidates, dealing a 5-card poker hand, or selecting lottery numbers.

Frequently Asked Questions

What is the value of 0! (Zero Factorial)?

By mathematical definition and gamma function convergence, \(0! = 1\). This ensures that \(\binom{n}{0} = 1\) and \(\binom{n}{n} = 1\) remain consistent.

Why are combinations always fewer than permutations (for r > 1)?

Because every combination group of size \(r\) can be arranged in \(r!\) different ways. Hence, \(nPr = nCr \times r!\).

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Permutations & Combinations Solver

RESULTS BREAKDOWN

📝 Step-by-Step Factorial Simplification

📊 Binomial Coefficient Distribution (n = 8)

Comprehensive Guide to Permutations, Combinations & Probability

When to Use Permutations vs. Combinations

Frequently Asked Questions

What is the value of 0! (Zero Factorial)?

Why are combinations always fewer than permutations (for r > 1)?

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