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Combinations Calculator

Calculate the number of possible combinations based on the entered data.

C(n, r) =n!
r! (n - r)!
Result:

The combinations calculator helps you determine the number of possible combinations that can be created by selecting items from a larger set. Whether you're organizing a team, planning a lottery, or solving combinatorial puzzles, this calculator makes it easy.

How to calculate combinations?

Combinations

To calculate combinations, divide the factorial of n by the product of the factorials of r and (n - r).

C(n, r) =n!
r! (n - r)!

Combinations with repetitions

To calculate combinations with repetitions, add the total number of distinct items (n) to the number of repetitions allowed (r) and subtract 1. Then, calculate the factorial of this sum and divide it by the product of the factorials of r and (n - r).

C(n + r - 1, r) =(n + r - 1)!
(r! * (n - r)!)

Examples

1. Select 6 Numbers from a Lottery Pool of 49 Numbers

You are playing a lottery game where you need to choose 6 numbers from a pool of 49 numbers. How many distinct combinations of 6 numbers can you select for your lottery ticket?

C(49,6) = 49! / (6! * (49-6)!) = 49! / (6! * 43!) = 13,983,816

Answer: There are 13,983,816 unique combinations of 6 numbers you can select for your lottery ticket.

2. Select 3 Books from a Shelf of 10 Books

You have a bookshelf with 10 different books, and you want to choose 3 books to take with you on a trip. How many unique combinations of 3 books can you select?

C(10,3) = 10! / (3! * (10-3)!) = 10! / (3! * 7!) = 120

Answer: There are 120 possible combinations of 3 books you can choose from the shelf.

3. Create a Team of 5 Players from a Pool of 12 Players

You're forming a sports team and need to select 5 players from a pool of 12 talented athletes. How many different combinations of 5 players can you choose for your team?

C(12,5) = 12! / (5! * (12-5)!) = 12! / (5! * 7!) = 792

Answer: There are 792 possible combinations of 5 players you can choose for your team.

Examples with repetitions

1. Select 3 Letters from the Alphabet (with repetitions allowed)

You want to form 3-letter strings using the English alphabet, allowing repetitions of letters. How many different combinations of 3 letters can you create?

C(3 + 26 - 1, 3) = C(28, 3) = 28! / (3! * (28-3)!) = 3,276

Answer: There are 3,276 unique combinations of 3 letters that can be formed with repetitions from the alphabet.

2. Pick 4 Dice Rolls (each roll can be any number from 1 to 6)

You are playing a dice game and want to determine the number of possible combinations when rolling 4 standard six-sided dice. How many different outcomes can you have?

C(4 + 6 - 1, 4) = C(9, 4) = 9! / (4! * (9-4)!) = 126

Answer: There are 126 unique combinations of rolling 4 dice, allowing each dice to show any number from 1 to 6.

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