Permutation Without Repetition
Calculate permutations without repetition (nPr) using n!/(n-r)!. Enter n objects and r selections to find total arrangements. Free online permutation calculator with charts.
About This Calculator
The Permutation Without Repetition Calculator computes the number of ways to arrange r items selected from a set of n distinct objects where the order matters and no object can be used more than once. This fundamental combinatorics concept is essential for ranking problems, scheduling, seating arrangements, competition placements, and any scenario involving ordered selections without replacement.
The calculation uses the formula P(n,r) = n! / (n-r)!, where n is the total number of distinct items and r is the number of selections to make. For example, choosing and arranging 3 books from a shelf of 10 distinct books gives P(10,3) = 10×9×8 = 720 possible arrangements. The formula works by multiplying n by (n-1) by (n-2) and so on until you have r factors, representing the decreasing number of choices at each step.
How It Works
Enter the total number of objects (n) and the sample size (r). The calculator instantly computes P(n,r) using the formula. It also computes the combination value C(n,r) and the ratio between them for comparison. The bar and line charts visualize how the number of permutations grows as r increases from 1 to the entered value, showing the characteristic steep growth curve of factorial-based combinatorics.
Regional Notes
India (IN): Permutations without repetition are a core topic in JEE Main and Advanced mathematics syllabus under permutations and combinations (P&C). Common problems involve arranging letters in words without repeating letters, forming numbers without digit repetition, and solving seating arrangement problems.
United States (US): Taught in AP Statistics and high school algebra under counting principles and probability. Used in real-world applications including race outcome predictions, lottery probability analysis (when order matters), and tournament bracket arrangements.
United Kingdom (UK): Covered in A-Level Mathematics under the probability and combinatorics curriculum. Applications include evaluating rankings in competitions, cryptographic key space analysis, and problem-solving in decision mathematics.
Frequently Asked Questions
What is permutation without repetition?
A permutation without repetition is an ordered arrangement of items where each item can be used at most once. For example, arranging 3 out of 5 books on a shelf: the number of ways is P(5,3) = 5×4×3 = 60. The order matters (ABC is different from CBA) and no item is repeated.
What is the formula for permutation without repetition?
The formula is P(n,r) = n! / (n-r)!, where n is the total number of distinct items and r is the number of items selected. For example, selecting 3 winners from 10 contestants in order (1st, 2nd, 3rd): P(10,3) = 10! / 7! = 10×9×8 = 720.
What is the difference between permutation with and without repetition?
Permutation without repetition (n!/(n-r)!) prohibits reusing items in arrangements, while permutation with repetition (n^r) allows items to be reused. For example, with n=10 digits and r=4 PIN digits: without repetition gives 10×9×8×7 = 5,040; with repetition gives 10^4 = 10,000.
How do you calculate permutations without repetition?
To calculate permutations without repetition, multiply n by (n-1) by (n-2) ... until you reach r factors. For example, P(8,3) = 8×7×6 = 336. Alternatively, compute n!/(n-r)! using the factorial function. For instance, arranging 4 books from 7: 7×6×5×4 = 840.
What are real-world examples of permutation without repetition?
Common examples include: assigning 1st/2nd/3rd place winners (order matters, no ties or repeats), arranging books on a shelf, forming distinct numbers without repeating digits, seating arrangements where each person occupies one chair, and lottery draws where order of drawn numbers matters.
What is the difference between permutation and combination without repetition?
In permutations without repetition, order matters: P(n,r) = n!/(n-r)!. In combinations without repetition, order does not matter: C(n,r) = n!/(r!(n-r)!). For example, choosing 3 winners from 10 with order (1st/2nd/3rd): P(10,3) = 720. Without order (just 3 winners): C(10,3) = 120. The ratio P/C is always r!.
Why is the number of permutations always larger than combinations?
Permutations count each ordering separately, so P(n,r) = C(n,r) × r!. Since r! ≥ 1, permutations are r! times larger than combinations. For example, P(10,3) = 720 while C(10,3) = 120 — exactly 6 = 3! times larger. This is because each group of r items has r! possible orderings.
What happens when r equals n in a permutation without repetition?
When r = n, P(n,n) = n!. This means arranging all n items in order. For example, the number of ways to arrange 6 books on a shelf is P(6,6) = 6! = 720. This is also known as the number of permutations of all n items, which simply equals n factorial.
Can permutation without repetition be applied to ranking problems?
Yes. Ranking problems use P(n,r) because order matters and no ties are allowed. For example, ranking 5 contestants (1st-5th) from a pool of 20: P(20,5) = 20×19×18×17×16 = 1,860,480 possible rankings. Race results, competition rankings, and award placements all use permutations without repetition.