What's the difference between Combination and Permutation?

**Combination (order doesn't matter)**: Selecting {A, B, C} is the same as {C, B, A}—only 1 combination. **Permutation (order matters)**: ABC, ACB, BAC, BCA, CAB, CBA are all different—6 permutations. Use combinations for lottery picks, team selection, card hands. Use permutations for passwords, race rankings, seating charts.

Is C(n, r) always smaller than P(n, r)?

Yes (except when r=0 or r=1, where they're equal). Permutations count every ordering as unique, while combinations treat all orderings as duplicates. The relationship is C(n,r) = P(n,r) / r!. For example: P(5,3) = 60 but C(5,3) = 10. The 60 permutations collapse into 10 combinations because each combination has 3! = 6 orderings.

If you select all items (n = r), there's only **1 combination**. For example, C(5,5) = 1 because there's only one way to select all 5 items—take everything. However, P(5,5) = 120 because there are 120 ways to arrange those 5 items.

How do I calculate lottery odds?

Use C(n, r) where n = total numbers and r = numbers picked. For a 6/49 lottery: C(49,6) = 13,983,816. Your odds of winning are 1 in 13,983,816. For Powerball (5 from 69 + 1 from 26): C(69,5) × C(26,1) = 292,201,338 combinations. This shows why lotteries are such long shots!

What's the connection to Pascal's Triangle?

Each number in Pascal's Triangle is a combination! Row n, position r gives C(n,r). For example, row 4 is: 1, 4, 6, 4, 1, which are C(4,0), C(4,1), C(4,2), C(4,3), C(4,4). The triangle's additive property (each number = sum of two above) reflects the formula C(n,r) = C(n-1,r-1) + C(n-1,r).

Can this calculate poker hand probabilities?

Yes! Total 5-card hands: C(52,5) = 2,598,960. Royal flush: C(4,1) = 4 ways (one per suit). Probability = 4/2,598,960 = 0.00015%. Four of a kind: C(13,1) × C(48,1) = 624 ways. Probability = 624/2,598,960 = 0.024%. Combinations are essential for all poker math!

Combination Calculator

Combination Calculator Overview

Calculate nCr combinations (Order doesn't matter)

The **Combination Calculator** is a combinatorics tool that calculates the number of ways to select items from a set where **order does NOT matter**. Whether you're calculating lottery odds, forming teams, analyzing card game probabilities, or solving mathematical problems, this calculator provides instant, accurate combination calculations for scenarios where sequence is irrelevant. **Combinations** are fundamental in probability theory, statistics, and decision-making. Unlike permutations where ABC and BAC are different, combinations treat them as identical—only the selection matters, not the arrangement. This concept is essential for understanding lottery mathematics, committee formation, card game odds, and sampling theory. ### The Combination Formula **C(n, r) = n! / (r! × (n - r)!)** Also written as: **ⁿCᵣ** or **(n choose r)** Where: - **n** = Total number of items in the set - **r** = Number of items to select - **!** = Factorial operation ### Why Combinations Are Smaller Than Permutations For the same n and r, C(n,r) ≤ P(n,r) because combinations eliminate duplicate orderings. Example with 3 items (A, B, C), choosing 2: - **Permutations**: AB, BA, AC, CA, BC, CB = 6 ways - **Combinations**: AB, AC, BC = 3 ways (BA is same as AB) The relationship: C(n,r) = P(n,r) / r! ### Real-World Applications **Lottery & Gambling:** - Calculate lottery winning odds - Determine poker hand probabilities - Analyze raffle ticket chances - Estimate casino game odds **Team & Committee Formation:** - Calculate possible team combinations - Determine committee selection options - Analyze group formation scenarios - Plan tournament pairings **Statistics & Sampling:** - Calculate sample selection possibilities - Determine survey group combinations - Analyze experimental design options - Study population sampling **Mathematics & Education:** - Solve combinatorics problems - Learn probability fundamentals - Understand binomial coefficients - Study Pascal's triangle ### Practical Examples **Example 1: Lottery Odds** Pick 6 numbers from 49 (standard lottery): - C(49, 6) = 49!/(6! × 43!) = **13,983,816 combinations** - Your odds of winning: 1 in 13,983,816 **Example 2: Team Selection** 10 players, need to choose 5 for starting lineup: - C(10, 5) = 10!/(5! × 5!) = **252 possible teams** **Example 3: Pizza Toppings** 12 toppings available, choose any 3: - C(12, 3) = 12!/(3! × 9!) = **220 combinations** **Example 4: Card Hand** How many 5-card poker hands from 52 cards? - C(52, 5) = 52!/(5! × 47!) = **2,598,960 hands**

How to Use Combination Calculator

**Identify Your Scenario**: Confirm that order doesn't matter. If selecting team members or lottery numbers (where {A,B,C} = {C,B,A}), use combinations.
**Enter Total Items (n)**: Input the total number of items available (e.g., 49 for lottery numbers 1-49).
**Enter Selection Size (r)**: Input how many items you're selecting (e.g., 6 for a 6-number lottery pick).
**Verify n ≥ r**: Ensure you're not selecting more items than available. The calculator validates this automatically.
**Click Calculate**: The tool computes C(n,r) using the factorial formula.
**Review the Result**: See the total number of unique combinations (selections without regard to order).
**Compare to Permutations**: Notice how C(n,r) is always ≤ P(n,r) because order-duplicates are eliminated.

Frequently Asked Questions

What's the difference between Combination and Permutation?: **Combination (order doesn't matter)**: Selecting {A, B, C} is the same as {C, B, A}—only 1 combination. **Permutation (order matters)**: ABC, ACB, BAC, BCA, CAB, CBA are all different—6 permutations. Use combinations for lottery picks, team selection, card hands. Use permutations for passwords, race rankings, seating charts.
Is C(n, r) always smaller than P(n, r)?: Yes (except when r=0 or r=1, where they're equal). Permutations count every ordering as unique, while combinations treat all orderings as duplicates. The relationship is C(n,r) = P(n,r) / r!. For example: P(5,3) = 60 but C(5,3) = 10. The 60 permutations collapse into 10 combinations because each combination has 3! = 6 orderings.
What if n equals r?: If you select all items (n = r), there's only **1 combination**. For example, C(5,5) = 1 because there's only one way to select all 5 items—take everything. However, P(5,5) = 120 because there are 120 ways to arrange those 5 items.
How do I calculate lottery odds?: Use C(n, r) where n = total numbers and r = numbers picked. For a 6/49 lottery: C(49,6) = 13,983,816. Your odds of winning are 1 in 13,983,816. For Powerball (5 from 69 + 1 from 26): C(69,5) × C(26,1) = 292,201,338 combinations. This shows why lotteries are such long shots!
What's the connection to Pascal's Triangle?: Each number in Pascal's Triangle is a combination! Row n, position r gives C(n,r). For example, row 4 is: 1, 4, 6, 4, 1, which are C(4,0), C(4,1), C(4,2), C(4,3), C(4,4). The triangle's additive property (each number = sum of two above) reflects the formula C(n,r) = C(n-1,r-1) + C(n-1,r).
Can this calculate poker hand probabilities?: Yes! Total 5-card hands: C(52,5) = 2,598,960. Royal flush: C(4,1) = 4 ways (one per suit). Probability = 4/2,598,960 = 0.00015%. Four of a kind: C(13,1) × C(48,1) = 624 ways. Probability = 624/2,598,960 = 0.024%. Combinations are essential for all poker math!

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