Calculate combinations (nCr) with step-by-step solutions, Pascal's Triangle visualization, and practical examples like lottery odds and team selection.
Combinations
120
C(10, 3)
Total number of items to choose from
Number of items to select
CR(n,r) = C(n+r-1, r)
Visual representation (limited to n <= 12)
Ways to select 3 items from 10 (order does not matter)
120
C(10, 3)
Combinations (order does not matter)
120
Permutations (order matters)
720
Ratio: P(10,3) / C(10,3) = 6 = 3!
C(10, 3) = 120 is highlighted at row 10, position 3
Property: Sum of row 10 = 1,024 = 210
Click to calculate:
Picking 3 numbers from 10
1 in 120
Probability: 0.83333333%
C(n, r) = n! / (r! x (n-r)!)
Order does not matter, no replacement
CR(n, r) = C(n+r-1, r)
Order does not matter, with replacement
C(n, r) = C(n, n-r)
C(10, 3) = C(10, 7)
C(n, r) = C(n-1, r-1) + C(n-1, r)
Each Pascal entry is sum of two above
Combinations
120
C(10, 3)
Combinations count selections where order does NOT matter. Formula: C(n,r) = n! / (r! x (n-r)!). For example, choosing 3 people from 10 for a committee: C(10,3) = 10!/(3! x 7!) = 120 ways. With replacement, use CR(n,r) = C(n+r-1, r). Combinations are used for lottery odds, team selection, and any scenario where arrangement order is irrelevant.
A combination is a selection of items from a collection where the order of selection does not matter. Unlike permutations, combinations treat ABC and CBA as the same selection. The combination formula C(n,r) = n!/(r!(n-r)!) counts the number of ways to choose r items from n items. Combinations are fundamental to probability theory, statistics, and are commonly used in lottery calculations, committee formation, and sampling problems.
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Combinations count selections where order does NOT matter. Formula: C(n,r) = n! / (r! x (n-r)!). For example, choosing 3 people from 10 for a committee: C(10,3) = 10!/(3! x 7!) = 120 ways. With replacement, use CR(n,r) = C(n+r-1, r). Combinations are used for lottery odds, team selection, and any scenario where arrangement order is irrelevant.
A combination is a selection of items where ORDER DOES NOT MATTER. C(n,r) counts ways to choose r items from n items. Example: Choosing 3 people from 10 for a committee = C(10,3) = 120 different groups.
Permutation: ORDER MATTERS (ABC differs from BAC). Use for rankings, arrangements. Combination: ORDER DOES NOT MATTER (ABC = BAC). Use for selections, committees. P(n,r) is always >= C(n,r).
Combinations with replacement allow selecting the same item multiple times. Formula: CR(n,r) = C(n+r-1, r). Example: Choosing 3 scoops of ice cream from 5 flavors (can repeat) = CR(5,3) = C(7,3) = 35.
Pascal's Triangle is a triangular array where each number is the sum of the two numbers above it. The entry at row n, position r equals C(n,r). It's useful for quickly finding combination values and has many mathematical properties.
Lottery odds use combinations because order does not matter. For a 6/49 lottery: C(49,6) = 49!/(6! x 43!) = 13,983,816 possible combinations. Your odds of winning are 1 in 13,983,816.
Last updated: 2025-01-15
Combinations
120
C(10, 3)