Permutation: Arrangements Where Order Matters
P(n,r) = n!/(n-r)! โ ways to arrange r items from n distinct items. Order matters: ABC โ ACB. With repetition: n^r. Circular: (r-1)!.
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P(n,r) = n!/(n-r)!. P(5,3)=5ร4ร3=60. With repetition: 3 digits from 0-9 = 10ยณ=1000. Circular: 5 people around table = (5-1)!=24.
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Why: Permutations count arrangements: passwords, race rankings, seating. Order matters. P(5,3)=60 ways to arrange 3 from 5. Combinations ignore order.
How: P(n,r)=nร(n-1)ร...ร(n-r+1)=n!/(n-r)!. r โค n. With repetition: n^r (each slot independent). Circular: fix one, arrange rest (r-1)!.
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๐ Step-by-Step Breakdown
For educational and informational purposes only. Verify with a qualified professional.
๐งฎ Fascinating Math Facts
P(n,r) = n!/(n-r)!
โ Arrangements
Order matters: ABC โ ACB
โ Permutation vs combination
๐ Key Takeaways
- โข P(n,r) = n!/(n-r)! โ arrangements of r items from n, order matters
- โข With repetition: n^r โ each position can be any of n items
- โข Circular: (r-1)! distinct arrangements around a circle (rotation = same)
- โข nP0 = 1, nPn = n!, nP1 = n
- โข Relation: P(n,r) = C(n,r) ร r! โ permutations = combinations ร ways to order
๐ก Did You Know?
๐ How It Works
P(n,r) counts ordered selections: first choice n options, second (n-1), third (n-2), โฆ, r-th (n-r+1). Product = n ร (n-1) ร โฆ ร (n-r+1) = n!/(n-r)!.
With repetition, each of r positions has n independent choices, so n^r. For circular permutations, fix one item and arrange the rest: (r-1)!.
๐ Worked Example: P(5,2)
Step 1: P(5,2) = 5! / (5-2)! = 5! / 3!
Step 2: 5! = 120, 3! = 6
Step 3: 120 / 6 = 20
Interpretation: 20 ordered pairs from 5: (1,2), (2,1), (1,3), (3,1), โฆ
๐ Real-World Applications
๐ Race Rankings
P(n,r) for top r finishers from n runners.
๐ PINs & Passwords
10^4 = 10,000 4-digit PINs (with repetition).
๐ License Plates
P(26,3) ร P(10,3) for letter-digit combos.
๐ญ Seating Arrangements
n! ways to seat n people in a row.
๐ฌ Experiment Order
Order of treatments in clinical trials.
๐ Rankings & Lists
Top-k rankings from n candidates.
โ ๏ธ Common Mistakes to Avoid
- Confusing P and C: Use P when order matters (ABC โ BAC); C when order doesn't matter.
- Using P when n^r applies: PINs allow repetition โ use n^r, not P(n,r).
- Circular vs linear: Circular divides by n: (n-1)! vs n! for linear.
- r > n: P(n,r) = 0 when r > n. Check constraints.
- Identical items: Use n!/(n1!รn2!รโฆ) when some items repeat.
๐ฏ Expert Tips
๐ก Falling Factorial
nPr = nร(n-1)รโฆร(n-r+1). More efficient than n!/(n-r)! for large n.
๐ก Order Matters
ABC vs BAC โ different permutations. Use P when sequence matters.
๐ก Circular Divide by n
n! linear arrangements; n rotations each โ (n-1)! distinct circular.
๐ก Identical Items
n!/(n1!รn2!รโฆ) when n1 of type 1, n2 of type 2, etc.
๐ Reference Table
| Type | Formula |
|---|---|
| P(n,r) | n! / (n-r)! |
| With repetition | n^r |
| Circular | (r-1)! |
| nP0 | 1 |
| nPn | n! |
| P = C ร r! | P(n,r) = C(n,r) ร r! |
๐ Quick Reference
๐ Practice Problems
โ FAQ
What is P(n,r)?
Number of ways to select and arrange r items from n distinct items. Order matters โ ABC and BAC are different.
Permutation vs combination?
Permutation: order matters. Combination: order doesn't. P(n,r) = C(n,r) ร r!.
When is repetition allowed?
Use n^r when each position can be any of n items (e.g., PINs, passwords).
What are circular permutations?
(r-1)! โ arrangements around a circle where rotation gives the same arrangement.
Why nP0 = 1?
One way to select and arrange zero items: do nothing (empty arrangement).
How many 4-digit PINs?
10^4 = 10,000 if digits can repeat. P(10,4) = 5,040 if no repetition.
When to use (n-1)!?
For circular arrangements of n distinct items โ fix one, arrange the rest.
๐ Summary
P(n,r) = n!/(n-r)! counts ordered arrangements of r items from n. Use n^r when repetition is allowed. Circular permutations use (r-1)!. Order matters for permutations; use combinations when it doesn't.
โ Verification Tip
Verify: P(n,r) = C(n,r) ร r!. Check nP0 = 1, nP1 = n, nPn = n!. For circular, (n-1)! ร n = n!.
๐ Next Steps
Explore the Combination Calculator for unordered selections, the Factorial Calculator for n!, or the Binomial Coefficient Calculator for Pascal's triangle.
โ ๏ธ Disclaimer: For n > 170, factorial overflow may occur. Use smaller values for exact results.
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