How many combinations in a 5 horse box trifecta? Sixty. Six in a 3 horse trifecta, 24 in a 4, 120 in a 6, and 210 in a 7. Those numbers aren’t negotiable — they’re the output of a permutation formula that applies identically whether you’re betting at Cheltenham or calculating in a spreadsheet. The formula for a box tricast is n multiplied by (n minus 1) multiplied by (n minus 2), where n is your number of selections. For a box forecast, it’s n multiplied by (n minus 1). For a superfecta, add a fourth factor: n multiplied by (n minus 1) multiplied by (n minus 2) multiplied by (n minus 3).
The tables below give you every number from 2 selections up to 10, for all three bet types, at multiple stake levels. Count before you box — the cost is always the combination count multiplied by your unit stake, and the difference between a four-horse box and a six-horse box is not a gentle increment. It’s a multiplication that turns a reasonable outlay into an expensive one faster than most people expect.
Box Exacta Permutations: 2 to 10 Horses
A box exacta — known in UK racing as a combination forecast — covers every possible first-and-second finishing order from your selected group. The formula is n times (n minus 1).
| Horses (n) | Formula | Combinations | Cost at £0.10 | Cost at £0.50 | Cost at £1 |
|---|---|---|---|---|---|
| 2 | 2 x 1 | 2 | £0.20 | £1.00 | £2 |
| 3 | 3 x 2 | 6 | £0.60 | £3.00 | £6 |
| 4 | 4 x 3 | 12 | £1.20 | £6.00 | £12 |
| 5 | 5 x 4 | 20 | £2.00 | £10.00 | £20 |
| 6 | 6 x 5 | 30 | £3.00 | £15.00 | £30 |
| 7 | 7 x 6 | 42 | £4.20 | £21.00 | £42 |
| 8 | 8 x 7 | 56 | £5.60 | £28.00 | £56 |
| 9 | 9 x 8 | 72 | £7.20 | £36.00 | £72 |
| 10 | 10 x 9 | 90 | £9.00 | £45.00 | £90 |
The growth rate is linear in the mathematical sense — each additional horse adds a constant increment to the multiplication. Going from 4 to 5 horses adds 8 combinations. Going from 5 to 6 adds 10. From 6 to 7 adds 12. The increments grow, but they grow steadily, not explosively. This makes box forecasts the most budget-friendly of the three exotic box types and the one where adding a marginal selection carries the lowest additional cost.
Box Trifecta Permutations: 3 to 10 Horses
A box trifecta — the combination tricast in UK terminology — covers every possible first-second-third finishing order. The formula is n times (n minus 1) times (n minus 2), and the cost curve steepens sharply compared to the forecast table.
| Horses (n) | Formula | Combinations | Cost at £0.10 | Cost at £0.50 | Cost at £1 |
|---|---|---|---|---|---|
| 3 | 3 x 2 x 1 | 6 | £0.60 | £3.00 | £6 |
| 4 | 4 x 3 x 2 | 24 | £2.40 | £12.00 | £24 |
| 5 | 5 x 4 x 3 | 60 | £6.00 | £30.00 | £60 |
| 6 | 6 x 5 x 4 | 120 | £12.00 | £60.00 | £120 |
| 7 | 7 x 6 x 5 | 210 | £21.00 | £105.00 | £210 |
| 8 | 8 x 7 x 6 | 336 | £33.60 | £168.00 | £336 |
| 9 | 9 x 8 x 7 | 504 | £50.40 | £252.00 | £504 |
| 10 | 10 x 9 x 8 | 720 | £72.00 | £360.00 | £720 |
The jump from 5 to 6 horses doubles the count: 60 to 120. From 6 to 7, it nearly doubles again: 120 to 210. This is the critical zone where most box bettors need to make a decision. Five horses at £1 costs £60 — a serious but manageable outlay for a Saturday handicap. Six horses at £1 costs £120, which is above most recreational budgets for a single bet. The 10p minimum stake softens this — six horses at 10p is £12 — but the proportional dividend also drops to a tenth. Every number in this table is a decision point, and the decision is always between breadth of coverage and depth of investment per line.
Box Superfecta Permutations: 4 to 10 Horses
The box superfecta — covering the first four finishing positions in any order — adds a fourth multiplicative factor and the costs escalate accordingly. The formula is n times (n minus 1) times (n minus 2) times (n minus 3).
| Horses (n) | Combinations | Cost at £0.10 | Cost at £1 |
|---|---|---|---|
| 4 | 24 | £2.40 | £24 |
| 5 | 120 | £12.00 | £120 |
| 6 | 360 | £36.00 | £360 |
| 7 | 840 | £84.00 | £840 |
| 8 | 1,680 | £168.00 | £1,680 |
| 9 | 3,024 | £302.40 | £3,024 |
| 10 | 5,040 | £504.00 | £5,040 |
At six horses and above, the superfecta enters territory that makes sense only at minimum stakes and only on the largest-field handicaps. A 7-horse box superfecta at 10p still costs £84. At £1 per line, you are looking at £840 — more than most punters would consider for a single race, regardless of the potential dividend. These are numbers to know rather than numbers to aim for — and they explain why the superfecta is the rarest standard exotic in UK racing.
Where These Numbers Come From: The Permutation Formula in Plain Terms
All three tables above are generated by the same underlying principle: ordered permutations. In mathematical notation, the number of ways to arrange r items chosen from a group of n is written as nPr, which equals n factorial divided by (n minus r) factorial. For box bets, r is 2 for forecasts, 3 for tricasts, and 4 for superfectas.
Factorial just means multiplying a number by every whole number below it: 5 factorial (written 5!) is 5 times 4 times 3 times 2 times 1, which equals 120. But the permutation formula simplifies in practice because the division cancels most of the lower terms. For a 5-horse box trifecta: 5P3 = 5! / (5-3)! = 5! / 2! = 120 / 2 = 60. You don’t actually need to compute factorials — just multiply the top r terms of n. Five horses, tricast: 5 times 4 times 3 equals 60. That’s the complete calculation.
The word “combination” in UK betting terminology is technically a misnomer in mathematical terms. In combinatorics, a combination is an unordered selection — where A-B-C is the same as C-B-A. A permutation is an ordered selection — where every rearrangement counts as a distinct outcome. What bookmakers call a “combination forecast” is actually a permutation-based bet, because the finishing order matters for the settlement of each individual line within the box. The box structure then covers all permutations so that the bettor doesn’t need to specify the order. The maths is permutational; the branding is combinatorial. It’s a harmless inconsistency, but it explains why the numbers in these tables grow the way they do.
In practical terms, you never need to calculate factorials at the racecourse. The shortcut is always the same: for a forecast, multiply the number of horses by one less than that number. For a tricast, multiply three consecutive descending numbers starting from your selection count. For a superfecta, four consecutive descending numbers. Bookmark the tables on this page, and the arithmetic takes care of itself. The only decision that matters is how many horses you’re willing to include — and the tables make the cost of that decision impossible to ignore.