Permutation Without Repetition Calculator - Count ordered selections of r from n
Permutation without repetition calculator for P(n, r) values with the supporting n! and (n - r)! factorials and the step-by-step multiplication chain.
Permutation Without Repetition Calculator
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What Is Permutation Without Repetition Calculator?
A permutation without repetition calculator returns P(n, r), the number of ordered arrangements of r items chosen from a set of n distinct items with no item used more than once. Type n and r and the tool returns P(n, r) alongside the supporting n! and (n - r)! factorials and the r-factor multiplication chain. P(n, r) = n! / (n - r)! is a classical counting rule in introductory statistics, and the calculator keeps the identity and its supporting factors visible in one read.
- • Counting race finishing orders: 8 runners competing for 3 podium positions gives P(8, 3) = 336 podiums with chain 8 * 7 * 6.
- • Counting PINs and codes with no repeats: A 4-character code from 10 digits with no repeated digit has P(10, 4) = 5040 codes (chain 10 * 9 * 8 * 7).
- • Counting batting orders: A coach picking an order from 9 players gets P(9, 9) = 9! = 362880 with chain 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1.
- • Checking counting-problem homework: Students plug in the same n and r and compare the calculator's P(n, r) against the n! / (n - r)! division they worked by hand.
P(n, r) is also written nPr, and the underlying identity is the same. This dedicated permutation without repetition calculator focuses on ordered arrangements with no item repeated. For the with-replacement case where the same item may appear in more than one position, the Permutation With Repetition Calculator uses n to the r instead of n! / (n - r)!.
For a tool that returns both P(n, r) and C(n, r) at once, the Permutation and Combination Calculator sits next to this one and shares the same inputs, so the reader can compare ordered and unordered counts for the same n and r.
How Permutation Without Repetition Calculator Works
The calculator reads n and r, validates that both are non-negative whole numbers with r <= n and n <= 20, then evaluates n! and (n - r)! using a guarded integer loop. P(n, r) is the integer quotient n! / (n - r)!. The same factorial products appear as a multiplication chain so the reader sees which r factors of n! survive the division.
- n (total items): Distinct items available to arrange. The calculator accepts whole numbers from 0 to 20.
- r (items arranged): Items to pick and arrange in order. Must satisfy 0 <= r <= n. r = 0 returns 1 and r = n returns n!.
- n! (n factorial): Product of every whole number from 1 to n. By definition 0! = 1, so the formula handles r = 0 without a special case.
- (n - r)! (denominator): Factorial of n - r. When r is small this factorial is tiny, and the division cancels a long tail of factors out of n!.
- Multiplication chain: The r largest factors of n! in descending order, equal to P(n, r) by cancellation.
The multiplication chain makes the cancellation visible. For P(8, 3) the chain is 8 * 7 * 6, which is 8! / 5!. The denominator drops 5, 4, 3, 2, 1 out of 8!, leaving the three largest factors.
Worked example: 5 items taken 3 at a time
n = 5, r = 3.
n! = 5 * 4 * 3 * 2 * 1 = 120. (n - r)! = (5 - 3)! = 2! = 2. P(5, 3) = 120 / 2 = 60. Multiplication chain: 5 * 4 * 3 = 60.
P(5, 3) = 60. n! = 120. (n - r)! = 2.
The 5! numerator has 5 factors; the 2! denominator cancels the 2 * 1 tail, leaving 5 * 4 * 3 = 60 ordered arrangements.
According to Wolfram MathWorld, the number of k-permutations of n objects is P(n, k) = n! / (n - k)!, valid for 0 <= k <= n.
Readers can pipe the factorial value into the Factor Calculator to list every whole-number factor of the input.
Key Concepts Explained
Four short definitions keep the formula honest. They hold whether the items are letters, numbers, or people.
Ordered arrangement
An arrangement is a sequence in which position matters. ABC and BAC are two different arrangements of the same three items, so both count separately toward P(n, r).
Factorial n!
The factorial of n multiplies every whole number from 1 up to n. 0! is defined as 1 by the empty-product convention, which lets the permutation formula handle r = 0 without a special case.
Cancellation of (n - r)!
Dividing n! by (n - r)! strips the smallest n - r factors out of the factorial. P(n, r) is the product of the r largest factors of n!, in descending order.
Permutation without repetition versus with repetition
Permutation without repetition forbids the same item from appearing twice. The count is n! / (n - r)!. Permutation with repetition allows repeats and uses n to the r instead.
These definitions matter because the same n and r can produce two different counts depending on whether the draw is with or without replacement. The Permutation With Repetition Calculator uses the with-replacement count n to the r, defined for every r >= 0; this calculator uses the no-replacement count n! / (n - r)!, defined only for 0 <= r <= n. The two formulas agree at r = 0 and r = 1, and they diverge elsewhere.
When the same item may appear in more than one position of the sequence, the Permutation With Repetition Calculator uses the related formula n to the r instead of n! / (n - r)! and is the right peer for that case.
How to Use This Calculator
Type n, type r, and read the result panel. The form validates the inputs and updates the result panel as you type.
- 1 Enter n: Type the size of the set in the Total Items (n) box. The default 5 covers the common counting example.
- 2 Enter r: Type the slots to fill in the Items to Arrange (r) box. The default 3 matches P(5, 3).
- 3 Read P(n, r): The first result row shows P(n, r) as an integer. The default input reads P(5, 3) = 60.
- 4 Read the supporting factorials: The next two rows show n! and (n - r)!. For P(5, 3) the panel reads n! = 120 and (n - r)! = 2.
- 5 Read the multiplication chain: The Multiplication Chain row shows the r largest factors of n! multiplied together. For P(5, 3) the chain is 5 * 4 * 3 = 60.
- 6 Fix validation errors: If r > n, either input is negative, or n > 20, the panel surfaces an inline error.
A coach picking a gold, silver, and bronze order from 8 runners enters n = 8 and r = 3 and reads P(8, 3) = 336 podiums. The Multiplication Chain row reads 8 * 7 * 6 = 336.
A reader sizing a 4-character code from 10 digits with no digit repeated can cross-check the P(10, 4) = 5040 figure against the Password Combination Calculator for a real password-style sample.
Benefits of Using This Calculator
- • P(n, r) in a single read: The calculator returns the no-replacement count as a single integer, so a homework step or slide can quote the answer without the factorial division.
- • Supporting factorials alongside the answer: n! and (n - r)! appear next to P(n, r), so the reader sees the two factorials that built the count.
- • Multiplication chain makes the cancellation visible: The chain row shows the r largest factors of n! in descending order.
- • Validation that catches counting problems early: If r > n, either input is negative, or n > 20, the form surfaces an inline error.
- • Exact integer results up to n = 20: The cap on n keeps 20! inside the safe-integer range, with no rounding noise.
Readers who need to turn P(n, r) into a probability can divide favorable outcomes by the sample space in the Probability Calculator.
Factors That Affect Your Results
Choice of n and r
P(n, r) grows quickly with both inputs. Adding one to n multiplies the count by the next whole number when r is unchanged; adding one to r multiplies by the next available factor from the top.
Whether items are distinct
The formula assumes every item is distinct. If the set has repeated items the count drops, because swapping two identical items does not produce a new arrangement.
Whether order matters
Permutations count ordered arrangements. If the downstream problem does not care about order, divide P(n, r) by r! to get C(n, r).
Whether repetition is allowed
When the same item may appear in more than one position, the count is n to the r rather than n! / (n - r)!. The no-replacement formula is undefined whenever r > n.
Upper bound of n = 20
The calculator caps n at 20 because 21! already exceeds the safe-integer range. The cap is a precision decision, not a counting decision.
- • The calculator assumes distinct items and no repetition. Problems with identical objects or with replacement need different formulas, such as n to the r.
- • The cap of n = 20 rules out large counting problems like P(52, 5) for 5-card poker. A big-integer library is the right tool for that range.
- • The result panel shows principal counts only, not a list of every arrangement.
According to OpenStax Introductory Statistics 2e, Section 3.1, the classical approach to probability counts favorable outcomes in the sample space, and that count for ordered arrangements is built from the permutation identity P(n, r) = n! / (n - r)!.
Once P(n, r) is the size of the ordered sample space, the Binomial Distribution Calculator treats it as the number of trials and turns a per-trial success rate into a probability distribution across r draws.
Frequently Asked Questions
Q: What is a permutation without repetition?
A: A permutation without repetition is an ordered arrangement of r items chosen from a set of n distinct items where no item is used more than once. The count is P(n, r) = n! / (n - r)!, and the calculator returns that count with the supporting factorials and the multiplication chain.
Q: What is the formula for permutation without repetition?
A: The formula is P(n, r) = n! / (n - r)!. The numerator multiplies every whole number from 1 to n; the denominator cancels the smallest n - r factors, leaving the r largest factors multiplied together. It is valid for whole numbers with 0 <= r <= n.
Q: How is permutation without repetition different from permutation with repetition?
A: Permutation without repetition forbids repeats and uses n! / (n - r)!, only defined for r <= n. Permutation with repetition allows the same item to repeat and uses n to the r. The two formulas agree at r = 0 and r = 1, and diverge elsewhere.
Q: How many permutations of 5 items taken 3 at a time?
A: P(5, 3) = 5! / (5 - 3)! = 120 / 2 = 60. The multiplication chain 5 * 4 * 3 = 60 is the shortcut. There are 60 ordered arrangements of 3 items chosen from a set of 5 distinct items.
Q: What is the difference between nPr and nCr?
A: nPr is the no-replacement ordered count, equal to n! / (n - r)!. nCr is the unordered count, equal to n! / (r! * (n - r)!). For the same n and r, nCr is always smaller than nPr because every combination of r items can be arranged in r! different orders.
Q: When is permutation without repetition undefined?
A: The formula is undefined when r > n, because (n - r)! has no factorial value for a negative input. The calculator surfaces an inline error for that case, so the user knows to either shrink r or switch to the with-replacement formula n to the r, which is defined for every r >= 0.