Combinations Calculator nCr Online
Find how many ways you can choose r items from n items where order does not matter
| Quantity | Value |
|---|
Combinations calculator results explain every value clearly
Combinations calculator results explain every value clearly so you understand exactly how the answer was reached. The table below shows common nCr values for quick reference when n is a small whole number.
| n | r | C(n, r) |
|---|---|---|
| 5 | 2 | 10 |
| 6 | 2 | 15 |
| 10 | 3 | 120 |
| 12 | 4 | 495 |
| 20 | 5 | 15504 |
| 25 | 3 | 2300 |
| 52 | 5 | 2598960 |
Combinations explained for everyday counting problems
Combinations explained for everyday counting problems means understanding that a combination counts groupings where the order of selection does not change the outcome. Picking a team of 3 players from a roster of 10 produces the same team regardless of the order players are chosen, so this is a combination problem rather than a permutation problem.
What n and r represent
What n and r represent is simple once you connect them to a real scenario. The value n is the total number of distinct items available, such as a deck of cards, a list of employees, or a menu of toppings. The value r is the size of the subset being chosen, such as a hand of 5 cards or a team of 3 employees.
Why order does not matter in combinations
Why order does not matter in combinations comes down to grouping versus arranging. Selecting prizes A then B is treated the same as selecting B then A because the resulting group of prizes is identical. This is the key distinction that separates combinations from permutations, where each different order counts as a separate outcome.
Common real world combination problems
Common real world combination problems include lottery number selections, committee formations, card hand probabilities, recipe ingredient choices, and sports team lineups. Any situation where you must select a group without regard to sequence can be solved using the nCr formula.
Combination formula derivation builds on factorials
Combination formula derivation builds on factorials because the total number of orderings of n items is n factorial, and dividing out the orderings within the chosen group and the remaining group removes duplicate counts. The formula C(n,r) = n! / (r!(n-r)!) is also called the binomial coefficient and is often written as n choose r.
Step by step example with n equals 6 and r equals 2
Step by step example with n equals 6 and r equals 2 starts by writing out the formula as 6! divided by (2! times 4!). Since 6! equals 720, 2! equals 2, and 4! equals 24, the calculation becomes 720 divided by 48, which equals 15. This means there are 15 different two item groups possible from a set of 6 items.
Handling zero and edge cases
Handling zero and edge cases requires knowing that 0 factorial equals 1 by definition. This means C(n,0) always equals 1 because there is exactly one way to choose nothing, and C(n,n) always equals 1 because there is exactly one way to choose everything. If r is greater than n, the combination is undefined and the result is 0.
Combinations versus permutations comparison clarifies usage
Combinations versus permutations comparison clarifies usage by focusing on whether sequence matters. Use a combination when the question asks how many groups, sets, or selections are possible. Use a permutation when the question asks how many ordered arrangements, rankings, or sequences are possible.
Quick test to identify the right formula
Quick test to identify the right formula is to ask whether swapping two chosen items creates a new outcome. If swapping creates a different result, the problem is a permutation. If swapping produces the same result, the problem is a combination, and the nCr formula applies.
Frequently asked questions about combinations nCr
A combination is a selection of items from a larger set where the order of selection does not matter. For example, choosing 2 toppings from a list of 5 is a combination because picking cheese then pepperoni is the same group as picking pepperoni then cheese.
The formula for nCr is n factorial divided by r factorial times the quantity n minus r factorial, written as C(n,r) = n! / (r!(n-r)!). This formula gives the total number of ways to choose r items from a set of n items without regard to order.
Combinations count selections where order does not matter, while permutations count arrangements where order does matter. Choosing 3 lottery numbers from a pool is a combination, but ranking 3 runners in first, second, and third place is a permutation.
No, r cannot be larger than n in a valid combination. The value of r must be between 0 and n inclusive, since you cannot choose more items than exist in the original set. If r equals 0 or r equals n, the result is always 1.
To calculate combinations by hand, multiply n by every whole number down to n minus r plus 1, then divide that product by r factorial. For example, 6 choose 2 equals (6 times 5) divided by (2 times 1), which equals 15.
0 factorial equals 1 by mathematical convention. This definition keeps the combinations formula consistent so that choosing 0 items from a set, or choosing all n items from a set, always equals exactly 1 way.