Combinatorics Calculator

Combinations Calculator

This calculator determines the number of ways to select y out of x items when the order in which they are selected does not matter.

Variable	Description	Value
x	Total number of items
y	Number of items chosen

Example	Value for x	Value for y
Determine the number of possible starting hands in a regular video poker game	52	5

Permutations Calculator

This calculator determines the number of ways to select y out of x items when the order in which they are selected does matter.

Variable	Description	Value
x	Total number of items
y	Number of items chosen

Example	Value for x	Value for y
Determine the number of unique possible B, I, G, or O columns on a Bingo card	15	5
Determine the number of unique possible N columns on a Bingo card	15	4
The total number of unique Bingo card permutations is therefore: Permut(15, 5) × Permut(15, 5) × Permut(15, 4) × Permut(15, 5) × Permut(15, 5) = 552,446,474,061,128,648,601,600,000

Hypergeometric Distribution Calculator

This calculator determines the probability of matching d out of c numbers when b numbers are randomly selected from a pool of a numbers.
This is useful for determining probabilities in Bingo, Keno and Lottery games.

Variable	Description	Value
a	Total number of items
b	Total number of items selected by the house
c	Total number of items selected by player
d	Total number of items needed to match

Example	Value for a	Value for b	Value for c	Value for d
Determine the odds of matching 4 out of 6 numbers in a traditional keno game	80	20	6	4
Determine the odds of matching 5 numbers in a lottery where 6 numbers from 1 to 50 are chosen	50	6	6	5
Determine the odds of having a full bingo card after 50 numbers are called	75	50	24	24

Practical Applications

Determining the odds of winning the Powerball Opens in a new window

can be done by using the Combinations Calculator:

Calculate the number of ways to select 5 of the 55 regular numbers = 3,478,761 combinations
Calculate the number of ways to select 1 of the 42 Powerball numbers = 42 combinations
Multiply the results together: 3,478,761 × 42 = 146,107,962.

There are a total of 146,107,962 ways to select (5 out of 55) and (1 out of 42) numbers. Only one of these combinations will win the jackpot. Therefore, the odds of winning the Powerball are 1 in 146,107,962.

To calculate the odds of matching 4 regular numbers without matching the Powerball, do the following:

Using the Hypergeometric Distribution Calculator, calculate the odds of matching 4 of 5 numbers when 5 of 55 are selected = 1 in 13,915.0440
Using the Combinations Calculator, determine the number of ways to select 41 out of 42 numbers = 42
Multiply the results together: 13,915.0440 × 42 = 584,431.8480

Therefore, the odds of matching 4 of 5 numbers without the Powerball are about 1 in 584,431.85.