Calculating Treble Chance Combinations by Hand, Step by Step

Digital coupon calculators have made working out perm sizes a non-event — type in your numbers, get your line count. But understanding the actual arithmetic behind that number is worth doing at least once, because it’s the difference between trusting a tool blindly and actually knowing why a bigger bank costs what it costs. Here’s how to work out “any 8 from n” combinations entirely by hand.

The Formula You’re Actually Using

The number of ways to choose 8 matches from a bank of n selections, where order doesn’t matter, is given by the combination formula:

C(n, 8) = n! / (8! × (n − 8)!)

Here, “!” means factorial — multiply the number by every whole number below it down to 1. It looks intimidating written out in full, but almost everything cancels before you need to multiply anything large, which is the trick to doing this by hand without a lot of tedious arithmetic.

Step 1: Write Out the Formula for Your Specific Bank Size

Say you’ve selected 10 matches and want every combination of 8. That’s C(10, 8) = 10! / (8! × 2!). Before multiplying anything out, notice that 10! contains 8! within it (10! = 10 × 9 × 8!), so the 8! on top and bottom cancel completely, leaving just (10 × 9) / 2!.

Step 2: Simplify Before You Multiply

2! is just 2 × 1 = 2. So the whole calculation collapses to (10 × 9) / 2 = 90 / 2 = 45. That’s your answer: an “any 8 from 10” perm produces 45 lines. No large factorials were ever actually needed — the cancellation did the heavy lifting.

Worked Example: 8 From 11

Follow the same shortcut for an 11-match bank. C(11, 8) = 11! / (8! × 3!). Again, 11! = 11 × 10 × 9 × 8!, so the 8! cancels, leaving (11 × 10 × 9) / 3!.

1. Multiply the top: 11 × 10 × 9 = 990.
2. Work out 3! on the bottom: 3 × 2 × 1 = 6.
3. Divide: 990 / 6 = 165.

So “any 8 from 11” produces 165 lines — exactly 3.67 times the 45 lines of the 10-match version, for the sake of covering just one additional selection.

Worked Example: 8 From 12

One more step up, using the same cancellation trick. C(12, 8) = 12! / (8! × 4!), which simplifies to (12 × 11 × 10 × 9) / 4!.

1. Multiply the top: 12 × 11 = 132, then 132 × 10 = 1,320, then 1,320 × 9 = 11,880.
2. Work out 4! on the bottom: 4 × 3 × 2 × 1 = 24.
3. Divide: 11,880 / 24 = 495.

An “any 8 from 12” perm produces 495 lines — exactly 11 times the 45-line version, and exactly 3 times the 165-line version.

The Shortcut Pattern Worth Remembering

Notice the recurring trick across all three examples: whenever you’re calculating C(n, 8), the 8! always cancels against the first eight terms of n!, leaving you with just the product of the “extra” terms above 8, divided by the factorial of however many extra selections you’ve added beyond 8. For a bank of n matches, that’s (n − 8)! on the bottom and the product of the top n − 8 descending numbers from n down to 9, give or take, on top. Once you see that pattern, you can build almost any “8 from n” perm size by hand without re-deriving the formula from scratch each time.

A Quick Way to Check Your Own Working

A useful sanity check: C(n, 8) should always equal C(n, n−8), because choosing which 8 matches to include is mathematically identical to choosing which n−8 matches to leave out. For the 11-match example, C(11, 8) = C(11, 3), and working out C(11, 3) directly — (11 × 10 × 9) / 3! — gives you exactly the same 990 / 6 = 165 you got the other way. If your two methods disagree, you’ve made an arithmetic slip somewhere and it’s worth rechecking the cancellation step.

Why Bother Doing This Manually At All

A coupon calculator will always be faster for day-to-day use, and there’s no need to recalculate your perm size by hand every single week. But understanding the mechanics means you’re never simply trusting a black box when deciding whether a bigger bank is worth the extra stake — you can see for yourself exactly why 11 matches costs 3.67 times as much as 10, and exactly how that multiple would change again if you stretched to 12 or 13 matches. That’s a genuinely useful piece of financial literacy for anyone staking real money on a perm week after week.

• Always cancel the 8! (or whichever smaller factorial matches your covered number) before multiplying anything large.
• Use the C(n, 8) = C(n, n−8) identity to double-check your arithmetic.
• Build a simple reference table for your own common bank sizes so you’re not redoing the full calculation every week.

Extending the Pattern to Larger Covered Numbers

Everything above covers “any 8 from n,” but the same cancellation trick applies if your coupon ever asks you to cover a different base number — say, any 6 from 9 for a smaller treble chance format. The formula is identical in structure: C(n, m) = n! / (m! × (n − m)!), and the same shortcut of cancelling the larger factorial against the matching portion of n! applies regardless of whether m is 6, 8, or any other covered number. Once you’ve worked through one or two examples by hand, the pattern becomes mechanical enough to apply to almost any perm size you encounter.

Closing Thought

None of this changes how likely any individual match is to draw — it’s purely the mechanics of how many lines a given perm size produces and what that costs. Knowing the arithmetic doesn’t give you an edge on the football itself, but it does mean every stake you commit to a bigger perm is a genuinely informed choice rather than a number taken on trust from an app.

Play the pools as a fixed-cost hobby and set your weekly budget before working out any perm size, large or small. If you’re 18 or over and choose to play, keep that budget steady regardless of how tempting a bigger bank looks once you’ve seen the maths behind it.