Actually there are 14,060 Group 6 winning combinations (the numerator of the fraction), not 579 (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 14,060 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(39,3) = 9139 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(39,3) = 182,780` such combinations.

Only some of those (3 in every 39 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(39,3) × 3/39 = 14,060` of these.

There are `8,145,060` possible combinations, so the fraction

`14060/8145060 = 1\ "in"\ 579`

is the chance of getting the Group 6 prize. The "579" does not refer to the number of possible combinations.

I hope that makes sense.