We can re-write the fraction as: `(185,115)/(13,983,816)xx3/43 = (12,915)/(13,983,816)`.

So actually there are 12,915 Group 6 winning combinations (the numerator of the final fraction), not 812 (which was obtained by cancelling).

For example, in the 19 Mar 2007 draw, the Winning Numbers were:

2 15 16 19 25 32

The Additional Number was 37

I won't list out all 12,915 combinations :-), but I will show how to get started.

Choosing 3 of the 6 winning numbers:

2 15 16
2 15 19
2 15 25
2 15 32

etc

There are 20 such combinations: `C(6,3)`

Now with each of these, we would have chosen 3 non-winning numbers, and there are C(43,3) = 12,341 such combinations for each of the 20 winning triples. For example, with the first 3 winning numbers, the non-winning combinations could include:

2 15 16 1 3 4
2 15 16 1 3 5
2 15 16 1 3 6
2 15 16 1 3 7
2 15 16 1 3 8
etc

There are `C(6,3) × C(43,3) = 246,820` such combinations.

Only some of those (3 in every 43 of them, in fact) will include the additional number 37, eg

2 15 16 1  3  37
2 15 16 1  37 38
2 15 16 37 38 39
etc

Other possible Group 6 combinations, using other winning numbers, are:

15 16 19 37 38 39
19 25 32 37 38 39
etc

There are `C(6,3) × C(43,3) × 3/43 = 17,220` of these.

There are `13,983,816` possible combinations, so the fraction

`(17,220)/(13,983,816) = 1\ "in"\ 812`

is the chance of getting the Group 6 prize. The "812" does not refer to the number of possible combinations.

I hope that makes sense.