Combinations Calculator

Combinations is evaluated from Total Items and Items to Choose. The calculation reports C - Combinations, P - Permutations and r!.

Results

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About the Combinations Calculator

Combinations is treated here as a quantitative relation between Total Items and Items to Choose and C - Combinations, P - Permutations, r! and Probability of Any 1 Combo.

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
C(n,r) = n! / (r! x (n - r)!)
P(n,r) = n! / (n - r)!
C(n,r) = P(n,r) / r!

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

C(n,r) = n! / (r! x (n - r)!)
P(n,r) = n! / (n - r)!
C(n,r) = P(n,r) / r!

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: Poker Hand — 5 Cards from 52

Inputs

n_val: 52 r_val: 5

C - Combinations: 2,598,960. P - Permutations: 311,875,200. r!: 120. Probability of Any 1 Combo: 0.0000003848

With Total Items = 52 and Items to Choose = 5 as the stated inputs, the result is C - Combinations = 2,598,960, P - Permutations = 311,875,200 and r! = 120. Each value corresponds to the declared output fields.

Example 2: Powerball Lottery Odds

Inputs

n_val: 69 r_val: 5

C - Combinations: 11,238,513. P - Permutations: 1,348,621,560. r!: 120. Probability of Any 1 Combo: 0.000000089

With Total Items = 69 and Items to Choose = 5 as the stated inputs, the result is C - Combinations = 11,238,513, P - Permutations = 1,348,621,560 and r! = 120. Each value corresponds to the declared output fields.

Example 3: Project Team Selection

Inputs

n_val: 15 r_val: 4

C - Combinations: 1,365. P - Permutations: 32,760. r!: 24. Probability of Any 1 Combo: 0.0007326007

With Total Items = 15 and Items to Choose = 4 as the stated inputs, the result is C - Combinations = 1,365, P - Permutations = 32,760 and r! = 24. Each value corresponds to the declared output fields.

Example 4: Coin Flips — Counting Heads

Inputs

n_val: 10 r_val: 3

C - Combinations: 120. P - Permutations: 720. r!: 6. Probability of Any 1 Combo: 0.0083333333

With Total Items = 10 and Items to Choose = 3 as the stated inputs, the result is C - Combinations = 120, P - Permutations = 720 and r! = 6. Each value corresponds to the declared output fields.

Common Use Cases

Calculate lottery odds (choosing 6 of 49 numbers)
Find number of possible card hands in poker
Count committee selection possibilities
Determine sample space size for probability