Permutation, in full
A permutation answers a counting question with order built in: how many ways can k items be pulled from n and lined up? The first slot has n candidates, the second has n-1 left, and so on down to n-k+1 for the last one. Multiplying those gives P(n,k) = n x (n-1) x ... x (n-k+1), written compactly as n! / (n-k)!. When k equals n the formula collapses to n!, the number of ways to arrange a whole set.
The contrast with a combination gets sharp once you use the same numbers for both. From 49 balls taking 6, P(49,6) = 49 x 48 x 47 x 46 x 45 x 44 = 10,068,347,520 ordered sequences, against C(49,6) = 13,983,816 unordered sets. The ratio between them is exactly 6! = 720, because any one set of six can be laid out in 720 orders. Standard draw games pay on the set, so the combination count is what sets the published odds.
Some games really do care about order. A straight play on a three-digit daily game asks for the digits in the exact positions drawn, while a boxed play on the same digits asks only for the set and pays less. Racing exactas and trifectas work the same way, settling on finishing order. Wherever a prize rule mentions position or sequence, the permutation count is the relevant one and the odds are longer than the equivalent unordered game.
- Probability
- Ordered arrangement, P(n,k)
A worked example
Three digits, 1, 2 and 3, taken all three at a time. P(3,3) = 3! = 6, namely 123, 132, 213, 231, 312 and 321. C(3,3) = 1, because all six of those are the same set. Scale it up and the gap widens fast: from 49 balls choosing 6, ordered counting gives 10,068,347,520 sequences against 13,983,816 sets, a factor of 720.
See also
Where it comes up on this site
Permutation: frequently asked
When the prize depends on position. A straight play on a three-digit daily game settles on the exact order drawn, so the outcome count is 1,000 rather than the smaller number of unordered sets.
By a factor of k!, because each unordered group of k items can be laid out in k! orders. With n = 49 and k = 6 the factor is 720.
Not in the main draw games. Daily digit games are the exception.