Combination Calculator - N Choose K - Online Number Generator

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Tool to generate combinations. In mathematics, a choice of k elements out of n distinguishable objects (k choose n), where the order does not matter, is represented by a list of elements, which cardinal is the binomial coefficient.

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Combination N Choose K - dCode

Tag(s) : Combinatorics

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Combination N Choose K

1. Mathematics
2. Combinatorics
3. Combination N Choose K

Combinations Generator

Choose (K) items

From the total number of items

Out of (N) Use digits (from 1 to N) Use letters (A,B,C…) Generate

From a custom list of items

Loading...(if this message do not disappear, try to refresh this page) Keep only distinguishable combinations (no duplicate) Generate See also: Permutations — Combinations with Repetition — Round-robin Tournament Generator — Covering Design for Lottery

Combinations with Order (1,2≠2,1)

⮞ Go to: K-Permutations — Permutations

Combinations with Repeated Items

⮞ Go to: Combinations with Repetition

Combinations Count Calculator

Choose (K) items Out of (N) Count See also: Binomial Coefficient

Combinations and Lottery Games

To get a list of combinations with a guaranteed minimum of numbers (also called reduced lottery draw), dCode has a tool for that:

⮞ Go to: Covering Design for Lottery

To draw random numbers (Lotto, Euromillions, Superlotto, etc.)

See also: Random Selection — Random Numbers

Answers to Questions (FAQ)

What is a combination of n choose k? (Definition)

A combination of $ k $ among $ n $ is the name given to the number of distinct ways of choosing $ k $ elements among a set of $ n $ distinct elements (with $ n \ge k $), without taking into account the order.

The combination is denoted by $ _nC^k $ or $ \binom{n}{k} $.

How to generate combinations of n choose k?

The generator allows selection of values $ k $ and $ n $, and generates possible lists of combinations with digits or letters (or a custom list).

Example: 4 choose 2 generates: (1,2),(1,3),(1,4),(2,3),(2,4),(3,4)

The generation is intentionally limited to a few thousand combinations, because the number of results grows very rapidly with $ n $ and $ k $. This limit prevents any server overload.

To generate larger lists, dCode can generate them upon (paid) request.

How to count the number of combinations of n choose k?

The calculation uses the binomial coefficient: $$ C_n^k = \binom{n}{k} = \frac{n!}{k!(n-k)!} $$

Combinations uses calculus of factorials (the exclamation mark: !).

3 choose 2 = 3 combinations	(1,2) (1,3) (2,3)
4 choose 2 = 6 combinations	(1,2) (1,3) (1,4) (2,3) (2,4) (3,4)
5 choose 2 = 10 combinations	(1,2) (1,3) (1,4) (1,5) (2,3) (2,4) (2,5) (3,4) (3,5) (4,5)
6 choose 2 = 15 combinations	(1,2) (1,3) (1,4) (1,5) (1,6) (2,3) (2,4) (2,5) (2,6) (3,4) (3,5) (3,6) (4,5) (4,6) (5,6)
7 choose 2 = 21 combinations	(1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (2,3) (2,4) (2,5) (2,6) (2,7) (3,4) (3,5) (3,6) (3,7) (4,5) (4,6) (4,7) (5,6) (5,7) (6,7)
8 choose 2 = 28 combinations	(1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (1,8) (2,3) (2,4) (2,5) (2,6) (2,7) (2,8) (3,4) (3,5) (3,6) (3,7) (3,8) (4,5) (4,6) (4,7) (4,8) (5,6) (5,7) (5,8) (6,7) (6,8) (7,8)
9 choose 2 = 36 combinations	(1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (1,8) (1,9) (2,3) (2,4) (2,5) (2,6) (2,7) (2,8) (2,9) (3,4) (3,5) (3,6) (3,7) (3,8) (3,9) (4,5) (4,6) (4,7) (4,8) (4,9) (5,6) (5,7) (5,8) (5,9) (6,7) (6,8) (6,9) (7,8) (7,9) (8,9)
4 choose 3 = 4 combinations	(1,2,3) (1,2,4) (1,3,4) (2,3,4)
5 choose 3 = 10 combinations	(1,2,3) (1,2,4) (1,2,5) (1,3,4) (1,3,5) (1,4,5) (2,3,4) (2,3,5) (2,4,5) (3,4,5)
6 choose 3 = 20 combinations	(1,2,3) (1,2,4) (1,2,5) (1,2,6) (1,3,4) (1,3,5) (1,3,6) (1,4,5) (1,4,6) (1,5,6) (2,3,4) (2,3,5) (2,3,6) (2,4,5) (2,4,6) (2,5,6) (3,4,5) (3,4,6) (3,5,6) (4,5,6)
7 choose 3 = 35 combinations	(1,2,3) (1,2,4) (1,2,5) (1,2,6) (1,2,7) (1,3,4) (1,3,5) (1,3,6) (1,3,7) (1,4,5) (1,4,6) (1,4,7) (1,5,6) (1,5,7) (1,6,7) (2,3,4) (2,3,5) (2,3,6) (2,3,7) (2,4,5) (2,4,6) (2,4,7) (2,5,6) (2,5,7) (2,6,7) (3,4,5) (3,4,6) (3,4,7) (3,5,6) (3,5,7) (3,6,7) (4,5,6) (4,5,7) (4,6,7) (5,6,7)
5 choose 4 = 5 combinations	(1,2,3,4) (1,2,3,5) (1,2,4,5) (1,3,4,5) (2,3,4,5)
6 choose 4 = 15 combinations	(1,2,3,4) (1,2,3,5) (1,2,3,6) (1,2,4,5) (1,2,4,6) (1,2,5,6) (1,3,4,5) (1,3,4,6) (1,3,5,6) (1,4,5,6) (2,3,4,5) (2,3,4,6) (2,3,5,6) (2,4,5,6) (3,4,5,6)
7 choose 4 = 35 combinations	(1,2,3,4) (1,2,3,5) (1,2,3,6) (1,2,3,7) (1,2,4,5) (1,2,4,6) (1,2,4,7) (1,2,5,6) (1,2,5,7) (1,2,6,7) (1,3,4,5) (1,3,4,6) (1,3,4,7) (1,3,5,6) (1,3,5,7) (1,3,6,7) (1,4,5,6) (1,4,5,7) (1,4,6,7) (1,5,6,7) (2,3,4,5) (2,3,4,6) (2,3,4,7) (2,3,5,6) (2,3,5,7) (2,3,6,7) (2,4,5,6) (2,4,5,7) (2,4,6,7) (2,5,6,7) (3,4,5,6) (3,4,5,7) (3,4,6,7) (3,5,6,7) (4,5,6,7)
6 choose 5 = 6 combinations	(1,2,3,4,5) (1,2,3,4,6) (1,2,3,5,6) (1,2,4,5,6) (1,3,4,5,6) (2,3,4,5,6)
7 choose 5 = 21 combinations	(1,2,3,4,5) (1,2,3,4,6) (1,2,3,4,7) (1,2,3,5,6) (1,2,3,5,7) (1,2,3,6,7) (1,2,4,5,6) (1,2,4,5,7) (1,2,4,6,7) (1,2,5,6,7) (1,3,4,5,6) (1,3,4,5,7) (1,3,4,6,7) (1,3,5,6,7) (1,4,5,6,7) (2,3,4,5,6) (2,3,4,5,7) (2,3,4,6,7) (2,3,5,6,7) (2,4,5,6,7) (3,4,5,6,7)

How to take into account the order of the elements?

By principle, combinations do not take into account order (1,2) = (2,1). Use the function permutations to get possible ordered combinations.

How to get combinations with repetitions?

dCode has a dedicated tool for combinations with repetitions.

How many combinations is there to lottery/euromillions?

To calculate the probabilities of winning in games of chance such as drawing random games, combinations are the most suitable tools.

To win at EuroMillions, a player ticks 5 boxes out of 50 (50 choose 5), then 2 stars out of 11 (11 choose 2).

Example: Calculate the number of combinations of (50 choose 5) = 2 118 760, and multiply by (11 choose 2) = 55 for a total of 116 531 800 combinations. The probability of winning is therefore 1 in 116 million.

To win at Powerball, pick 5 out of 69 (69 choose 5), then pick 1 out of 26 (26 choose 1).

Example: Calculate the number of combinations of (69 choose 5) = 11 238 513, and multiply by (26 choose 1) = 26 for a total of 292 201 338 combinations. The probability of winning is therefore 1 in 292 million.

To win at EuroDreams, the draw is 6 numbers from 40, then 1 number from 5.

Example: Calculate the number of combinations of (40 choose 6) = 3 838 380, and multiply by (1 among 5) = 5, for a total of 19 191 900 combinations. The probability of winning is therefore 1 chance in 19 million.

Many books describes strategies for lotto or lottery such as here (affiliate link) One of the strategies is to play covering designs systems.

Why k cannot be equal to zero 0?

If $ k = 0 $, no element is selected. There is then only one possible combination: the empty set. By combinatorial convention, $$ \binom{n}{0} = 1 $$

Why n cannot be equal to zero 0?

If $ n = 0 $, then there is 0 item, impossible to pick $ k $, so there are no results. So $$ \binom{0}{k} = 0 $$

What is the value of 0 choose 0?

By convention 0 choose 0 is 1: $$ \binom{0}{0} = 1 $$

What is the algorithm for counting combinations?

To count the combinations: // Pythondef count_combinations(n, k): if k > n - k: k = n - k result = 1 for i in range(1, k + 1): result = result * (n - i + 1) // i return result

What is the algorithm to generate combinations?

To list the combinations: // Pythondef combinations(n, k): result = [] combo = list(range(k)) while True: result.append(combo[:]) i = k - 1 while i >= 0 and combo[i] == n - k + i: i -= 1 if i < 0: break combo[i] += 1 for j in range(i + 1, k): combo[j] = combo[j - 1] + 1 return result// JavaScriptfunction combinations(a) { // a = new Array(1,2) var fn = function(n, src, got, all) { if (n == 0) { if (got.length > 0) { all[all.length] = got; } return; } for (var j = 0; j < src.length; j++) { fn(n - 1, src.slice(j + 1), got.concat([src[j]]), all); } return; } var all = []; for (var i=0; i < a.length; i++) { fn(i, a, [], all); } all.push(a); return all;}

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Summary

• Combinations Generator
• Combinations with Order (1,2≠2,1)
• Combinations with Repeated Items
• Combinations Count Calculator
• Combinations and Lottery Games
• What is a combination of n choose k? (Definition)
• How to generate combinations of n choose k?
• How to count the number of combinations of n choose k?
• How to take into account the order of the elements?
• How to get combinations with repetitions?
• How many combinations is there to lottery/euromillions?
• Why k cannot be equal to zero 0?
• Why n cannot be equal to zero 0?
• What is the value of 0 choose 0?
• What is the algorithm for counting combinations?
• What is the algorithm to generate combinations?

Similar pages

• Covering Design for Lottery
• Combinations with Repetition
• Permutations
• Round-robin Tournament Generator
• K-Permutations
• Binomial Coefficient
• Random Numbers
• DCODE'S TOOLS LIST

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