Combination & Permutation Calculator (nCr & nPr)
Free nCr and nPr calculator. Find combinations (order doesn't matter) and permutations (order matters) with step-by-step explanation.
nCr & nPr Calculator
(order doesn't matter)
(order matters)
Comprehensive Guide: Combinatorics, Permutations & Combinations (nPr & nCr)
Learn the fundamental counting principle, factorial expansions, order significance, and lottery probability calculations.
1. Fundamental Distinction: Order Matters vs. Order Does Not Matter
In combinatorics, selecting subsets of r items from a total pool of n distinct elements depends fundamentally on whether the sequence or order of selection matters:
- Permutation (nPr — Order Matters): Sequences where arrangement creates distinct outcomes. Podiums in a race (1st, 2nd, 3rd), PIN codes, and passwords are permutations.
{A, B} ≠ {B, A}. - Combination (nCr — Order Does NOT Matter): Groupings where internal arrangement is irrelevant. Selecting a committee of 3 people or a poker hand of 5 cards are combinations.
{A, B} = {B, A}.
2. Mathematical Formulas & Factorial Derivations
Combination: ⁿCᵣ = n! ÷ [r! × (n − r)!] = ⁿPᵣ ÷ r!
Notice that ⁿCᵣ is simply ⁿPᵣ divided by r!. Dividing by r! eliminates all duplicate internal permutations of the selected subgroup.
3. Graduated Step-by-Step Worked Problems
Problem 1: Selecting a Committee of 3 from 10 Candidates (Combination)
How many ways can a 3-person steering committee be selected from 10 volunteers?
Step 1: Identify parameters: n = 10, r = 3 (order does not matter).
Step 2: Apply Combination formula:
¹⁰C₃ = 10! / [3! × (10 − 3)!] = 10! / (3! × 7!).
Step 3: Expand and cancel common factorials:
(10 × 9 × 8 × 7!) / [(3 × 2 × 1) × 7!] = (10 × 9 × 8) / (3 × 2 × 1).
= 720 / 6 = 120 ways.
Result: There are exactly 120 possible committees.
Problem 2: Electing President, Vice President & Treasurer (Permutation)
If the same 10 candidates are electing 3 specific ranked officer positions:
• Order matters: ¹⁰P₃ = 10! / (10 − 3)! = 10! / 7! = 10 × 9 × 8 = 720 ways.
There are 6 times as many permutations because each 3-person committee can hold officers in 3! = 6 different permutations.
Problem 3: Circular Table Seating Permutations
In how many unique ways can 6 delegates sit around a circular conference table?
• Linear seating yields 6! = 720.
• Around a circle without fixed chairs, rotating all guests preserves relative position. We divide by 6 rotations:
Circular Permutations = (n − 1)! = (6 − 1)! = 5! = 120 ways.
Result: Exactly 120 circular arrangements exist.
4. Symmetry Property of Combinations: nCr = nC(n−r)
A powerful algebraic shortcut is the complementary symmetry property: ⁿCᵣ = ⁿCₙ₋ᵣ. For example, choosing 8 items out of 10 (¹⁰C₈) is mathematically identical to choosing which 2 items to leave behind (¹⁰C₂ = 45), dramatically speeding up hand calculations.
5. Frequently Asked Questions (FAQ)
Why is a 'combination lock' actually a misnomer?
A padlock requiring codes like 15-24-33 is mathematically a permutation lock. If it were a true combination lock, dialing 24-15-33 or 33-24-15 would still unlock it!
How do permutations work when elements repeat (like letters in 'MISSISSIPPI')?
For n total items where item 1 repeats p times, item 2 repeats q times: Permutations = n! / (p! · q! · ...). In MISSISSIPPI (11 letters: 1 M, 4 I, 4 S, 2 P): 11! / (4! · 4! · 2!) = 34,650 permutations.
What is the probability of winning a 6/49 lottery?
Selecting 6 numbers from 49: ⁴⁹C₆ = 49! / (6! · 43!) = 13,983,816 combinations. The probability of matching all 6 numbers with one ticket is exactly 1 in 13,983,816 (approx. 0.00000715%).