Permutation & Combination

1. **Enter n** — type the total number of items in your set (e.g., 10 students, 52 cards).
2. **Enter r** — type how many items you are selecting or arranging at a time (r must be ≤ n).
3. **Choose the operation** — select *Permutation (nPr)* if order matters (e.g., ranking, passwords) or *Combination (nCr)* if order does not matter (e.g., choosing a committee).
4. **Click Calculate** — the calculator applies the appropriate factorial formula and displays the exact result instantly.
5. **Interpret the result** — the number shown is the total count of distinct arrangements (permutation) or selections (combination) possible.

P(n,r) = n! / (n−r)!    |    C(n,r) = n! / [r! × (n−r)!]

• n — The total number of distinct items in the set (must be a non-negative integer).
• r — The number of items being chosen or arranged at a time (must satisfy 0 ≤ r ≤ n).
• n! — Factorial of n — the product of all positive integers from 1 up to n (e.g., 5! = 120). By convention, 0! = 1.
• (n − r)! — Factorial of the difference between n and r; appears in both formulas to cancel out items not selected.
• r! — Factorial of r; appears only in the combination formula to remove duplicate orderings of the selected items.
• P(n,r) — The number of permutations — distinct ordered arrangements of r items chosen from n items.
• C(n,r) — The number of combinations — distinct unordered subsets of r items chosen from n items.

**Permutation P(8,3):**
P(8,3) = 8! / (8−3)! = 8! / 5! = (8 × 7 × 6 × 5!) / 5! = 8 × 7 × 6 = **336**

**Combination C(8,3):**
C(8,3) = 8! / [3! × (8−3)!] = 8! / (6 × 120) = (8 × 7 × 6) / (3 × 2 × 1) = 336 / 6 = **56**

Result: P(8,3) = 336 ordered arrangements; C(8,3) = 56 unordered selections

With n = 8 and r = 3:

• 336 permutations means there are 336 different ways to pick and arrange 3 items in order from a group of 8. For example, awarding a 1st, 2nd, and 3rd place prize to 3 people from 8 contestants yields 336 distinct outcomes.
• 56 combinations means there are only 56 distinct groups of 3 items when order is irrelevant. For example, choosing a 3-person committee from 8 people gives 56 possible committees.

The ratio 336 / 56 = 6 = 3! confirms that each combination corresponds to exactly 3! = 6 different ordered permutations.

Permutations vs. Combinations — Key Concepts

When Does Order Matter?

The fundamental question is: does the sequence of selection change the outcome?

• Yes → use a Permutation. Examples: PIN codes, race finishing positions, letter arrangements in a word.
• No → use a Combination. Examples: lottery ticket numbers, pizza toppings, team rosters.

The Role of Factorials

Both formulas are built on factorials. The factorial n! grows extremely fast:

• 5! = 120
• 10! = 3,628,800
• 20! ≈ 2.43 × 10¹⁸

For large n and r, calculators or software are essential because hand computation becomes impractical.

Relationship Between nPr and nCr

nPr = nCr × r!

This shows that a permutation is simply a combination multiplied by the number of ways to order the chosen items (r!). Every combination can generate r! permutations.

Special Cases

• r = 0: C(n,0) = P(n,0) = 1 (there is exactly one way to choose or arrange nothing).
• r = n: C(n,n) = 1 (only one way to select all items); P(n,n) = n! (all possible orderings of the full set).
• r = 1: C(n,1) = P(n,1) = n.

Real-World Applications

| Scenario | Type | Why | |---|---|---| | Arranging books on a shelf | Permutation | Order of books changes the arrangement | | Choosing a lottery ticket | Combination | Numbers drawn in any order win | | Creating a password | Permutation | Different sequences are different passwords | | Forming a sports team | Combination | Team composition, not order, matters | | Assigning ranked medals | Permutation | Gold/Silver/Bronze are distinct positions |

• **Confusing permutation with combination** — always ask 'does order matter?' before choosing the formula. A common error is using nPr when selecting a committee (order irrelevant) instead of nCr.
• **Setting r > n** — both formulas are undefined when r exceeds n, because you cannot arrange more items than exist. Always verify r ≤ n before calculating.
• **Forgetting that 0! = 1** — when r = n, the formula includes (n−n)! = 0! = 1, not 0. Treating 0! as 0 makes the denominator zero and produces an error.
• **Misidentifying n and r** — n is always the larger set size; r is the subset size. Swapping them gives a completely wrong answer.
• **Ignoring identical items** — the standard nPr formula assumes all n items are distinct. If some items are identical (e.g., repeated letters), you must use the multinomial formula instead.
• **Calculator mode errors** — ensure your scientific calculator or spreadsheet function uses the correct function: PERMUT(n,r) for permutations and COMBIN(n,r) for combinations in Excel/Google Sheets.

• Combinatorics — Permutations and Combinations — Khan Academy
• Permutations and Combinations (NCERT Class 11 Mathematics, Chapter 7) — National Council of Educational Research and Training (NCERT), Government of India
• Combinatorics — Discrete Mathematics — Old Dominion University, Department of Computer Science
• COMBIN function — Microsoft Support — Microsoft Corporation
• Concrete Mathematics: A Foundation for Computer Science (2nd ed.), Graham, Knuth, Patashnik — Addison-Wesley / Pearson Education

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