Combination, in full
Combinatorics counts selections in two ways, and the split turns on whether order carries information. A combination throws order away. Picking 6, 12, 23, 31, 40 and 44 on a lottery slip is the same ticket as picking 44, 40, 31, 23, 12 and 6, so the two count once between them. The standard notation is C(n,k), also written as n choose k or as a two-row bracket, and it asks: from n distinct items, how many different unordered groups of size k exist?
The formula is C(n,k) = n! / (k!(n-k)!). The numerator n! counts every ordering of all n items. Dividing by (n-k)! ignores how the items you did not pick are arranged, which leaves P(n,k), the count of ordered selections. Dividing again by k! collapses the k! different orderings of any one chosen group down to a single entry. That second division is the entire difference between a combination and a permutation.
Knowing the count does not make any particular set likelier. C(49,6) = 13,983,816 says only that a 6-from-49 draw has that many equally likely outcomes, so every ticket carries the same 1-in-13,983,816 chance. Sets that look patterned, 1-2-3-4-5-6 among them, sit in the same pool as sets that look scattered. The one practical consequence of all this is prize sharing: sets that look appealing to humans get chosen by more people, so a win on one is split more ways.
- Probability
- n choose k, Binomial coefficient
A worked example
Take the 6-from-49 matrix. C(49,6) = 49! / (6! x 43!), which cancels down to (49 x 48 x 47 x 46 x 45 x 44) / (6 x 5 x 4 x 3 x 2 x 1). The numerator is 10,068,347,520 and the denominator is 720, so C(49,6) = 13,983,816. That is the jackpot count behind a 6-from-49 game such as Canada's Lotto 6/49.
Sources
A combination is the number of ways of picking k unordered outcomes from n possibilities, counted by n! / (k!(n-k)!).
Wolfram MathWorld — CombinationCanada's Lotto 6/49 draws six numbers from 49 and publishes jackpot odds of 1 in 13,983,816.
Wikipedia — Lotto 6/49
See also
Where it comes up on this site
Combination: frequently asked
13,983,816. That is C(49,6) = 49! / (6! x 43!), which reduces to 10,068,347,520 divided by 720. Every one of those sets is equally likely on a fair draw, so a single line carries a 1 in 13,983,816 chance of matching all six.
No. It is one set among 13,983,816 and carries the same chance as the rest. The pattern is visible to people and not to the machine. What it does have is company. Enough players pick it that a jackpot on those numbers would be split many ways.
Order. A permutation counts arrangements, a combination counts groups. C(n,k) = P(n,k) / k!.