Combination Calculator

Calculate combinations C(n, r) — the number of ways to choose r items from n without regard to order.

C(n,r)
120
10 choose 3
C(n,r) no rep
120
n! / (r!(n-r)!) = 10! / (3!7!)
P(n,r) no rep
720
n! / (n-r)! = 10! / 7!
C(n,r) with rep
220
(n+r-1)! / (r!(n-1)!)
P(n,r) with rep
1,000
n^r = 10^3
1/C(n,r) Probability
0.008333
Chance of selecting one specific combination
n!
3,628,800
Factorial of 10
r!
6
Factorial of 3

Nearby Values for n = 10

rC(10, r)P(10, r)Ratio P/C
0111
110101
245902
3 (current)1207206
42105,04024
525230,240120
6210151,200720

Pascal Triangle Row 10

k=0
1
k=1
10
k=2
45
k=3
120
k=4
210
k=5
252
k=6
210
k=7
120
k=8
45
k=9
10
k=10
1

Formula Reference

TypeFormulaDescription
CombinationC(n,r) = n! / (r!(n-r)!)Order does not matter, no repetition
PermutationP(n,r) = n! / (n-r)!Order matters, no repetition
Combination (rep)C(n+r-1, r)Order does not matter, with repetition
Permutation (rep)n^rOrder matters, with repetition
Planning notes, formulas, and examples

About the Combination Calculator

The Combination Calculator computes C(n, r), also written as "n choose r" or ⁿCᵣ. Combinations count the number of ways to select r items from a set of n items when the order of selection does not matter.

Combinations are fundamental in probability, statistics, and combinatorics. They answer questions like "How many 5-card poker hands can be dealt from a 52-card deck?" or "How many ways can a committee of 3 be formed from 10 people?"

This calculator computes the exact result using the formula C(n,r) = n! / (r!(n−r)!) and handles large numbers efficiently by simplifying before multiplying to avoid arithmetic overflow.

When This Page Helps

Computing combinations by hand requires factorial division that quickly overflows. This calculator handles the simplification for large values of n and r.

How to Use the Inputs

  1. Enter n (total number of items).
  2. Enter r (number of items to choose).
  3. The result is C(n, r) = n! / (r!(n−r)!).
  4. r must be ≤ n and both must be non-negative integers.
  5. Compare with the permutation count to see the effect of order.
Formula used
C(n, r) = n! / (r! × (n−r)!) Where: - n = total items - r = items chosen - ! = factorial

Example Calculation

Result: 120

C(10,3) = 10! / (3! × 7!) = (10×9×8) / (3×2×1) = 720/6 = 120. There are 120 ways to choose 3 items from 10.

Tips & Best Practices

  • C(n, 0) = C(n, n) = 1 — there's one way to choose nothing or everything.
  • C(n, r) = C(n, n−r) — choosing r to include is the same as choosing n−r to exclude.
  • The sum of all C(n, r) for r = 0 to n equals 2ⁿ.
  • Pascal's triangle gives combinations: row n, position r.
  • Lottery odds are calculated using combinations.
  • C(52, 5) = 2,598,960 — the number of possible 5-card poker hands.

Combinations in Probability

The probability of a specific combination outcome is 1 / C(n, r). This underpins lottery odds, card game probabilities, and sampling theory.

Combinations with Repetition

When items can be chosen more than once, the formula is C(n+r−1, r). For example, choosing 3 scoops from 5 flavors with repetition gives C(7, 3) = 35.

Computational Tricks

To avoid overflow, compute C(n, r) by multiplying r terms of n×(n−1)×...×(n−r+1) and dividing by r! simultaneously, or use the identity C(n, r) = C(n, n−r) to pick the smaller denominator.

Mastering this concept provides a strong foundation for advanced coursework in mathematics, statistics, and related quantitative disciplines.

Sources & Methodology

Last updated:

Frequently Asked Questions

  • A combination is a selection of items where order does not matter. Choosing {A, B, C} is the same as {C, B, A}. The count of such selections is given by C(n, r).

More in this topic

Permutation Calculator

Calculate permutations P(n, r) — the number of ordered arrangements of r items from n.

Factorial Calculator

Calculate the factorial of any non-negative integer (n!). See step-by-step multiplication and related values.