Calculate combinations C(n, r) — the number of ways to choose r items from n without regard to order. Free online nCr 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 tool 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.
Tracking this metric consistently enables professionals to identify patterns in how they allocate time and effort, revealing opportunities to work more effectively and accomplish more each day. This measurement provides a critical foundation for goal setting and progress tracking, helping you align daily activities with longer-term objectives and meaningful milestones.
Computing combinations by hand requires factorial division that quickly overflows. This calculator handles the simplification for large values of n and r. Data-driven tracking enables proactive schedule management, helping professionals protect focused work time and reduce the cognitive overhead of constant task-switching throughout the day. This quantitative approach replaces vague time estimates with concrete data, enabling professionals to plan realistic schedules and avoid the pattern of chronic overcommitment.
C(n, r) = n! / (r! × (n−r)!) Where: - n = total items - r = items chosen - ! = factorial
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.
The probability of a specific combination outcome is 1 / C(n, r). This underpins lottery odds, card game probabilities, and sampling theory.
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.
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.
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).
Combinations ignore order; permutations consider order. C(5,3) = 10 but P(5,3) = 60. Each combination corresponds to r! permutations.
C(n, r) is also called the binomial coefficient because it appears in the expansion of (a+b)ⁿ. It is a cornerstone of probability theory and combinatorics.
Lottery odds are 1 / C(n, r). For a 6/49 lottery, the odds are 1 / C(49, 6) = 1 / 13,983,816 ≈ 1 in 14 million.
Pascal's triangle arranges binomial coefficients in a triangle where each entry is the sum of the two entries above it. Row n gives all C(n, r) values.
No. You cannot choose more items than are available. C(n, r) = 0 when r > n by convention.