How many parlays are in a round robin by 2s, 3s and 4s

A round robin of n selections "by k" places every possible parlay of k legs from the n. The number is n choose k: n! ÷ (k! × (n − k)!). Three selections by 2s is 3 parlays, four by 2s is 6, five by 3s is 10, six by 4s is 15. The total stake is that count times the stake on each parlay.

The round robin calculator counts the parlays and the total stake for any list.

The formula, and a quick way to do it

For 5 selections by 3s, multiply the first 3 terms counting down from 5 and divide by 3 factorial: (5 × 4 × 3) ÷ (3 × 2 × 1) = 60 ÷ 6 = 10. For 6 by 4s: (6 × 5 × 4 × 3) ÷ (4 × 3 × 2 × 1) = 360 ÷ 24 = 15. The count of parlays by k is the same as by n − k, because choosing which legs to include is the same as choosing which to leave out.

The counts

• 3 selections: by 2s = 3
• 4 selections: by 2s = 6, by 3s = 4
• 5 selections: by 2s = 10, by 3s = 10, by 4s = 5
• 6 selections: by 2s = 15, by 3s = 20, by 4s = 15, by 5s = 6
• 7 selections: by 2s = 21, by 3s = 35, by 4s = 35
• 8 selections: by 2s = 28, by 3s = 56, by 4s = 70

The counts rise fast. Eight selections by 4s is 70 separate slips, and with a 10 stake that is a 700 outlay. People who add one more selection to a round robin are often surprised by how much the total stake grows.

How many parlays each leg appears in

Each selection appears in C(n − 1, k − 1) of the parlays. In five by 3s that is C(4,2) = 6. In six by 2s it is C(5,1) = 5. This number shows how much damage one losing selection does. In six by 2s, one loser kills 5 of 15 doubles. In six by 4s, a single loser kills C(5,3) = 10 of 15 parlays.

Several sizes at once

Many forms let you choose "by 2s and 3s" together. The counts add. Four selections by 2s and 3s is 6 + 4 = 10 parlays, and the stake is entered per parlay, so the outlay is 10 times the stake. Full cover bets in the UK style, such as a Lucky 15 on four selections, add the singles and the full accumulator added to the smaller groups, and forms differ on what they call them, so read the slip before relying on the name.

Two details that change the count

• Legs from the same game. Two selections from one game cannot be combined in a standard parlay, since their results are not independent. The round robin skips those combinations, and the count is below n choose k. Check the slip total before placing.
• Stake per parlay or total stake. Some slips ask for the stake on each parlay and some for the total to be spread across them. The difference is a factor equal to the count, so a 10 entered on the wrong line can mean a 10 or a 100 outlay.

How the structure pays when some legs lose is worked in the payout article, and the round robin overview covers what you are placing.

Count yours. Put the number of selections and the size into the round robin calculator to see the parlay count and the total outlay before you place anything.