Since 6 doesn't go exactly into 45, we need to take the nearest multiple of 6 which is lower than 45. 6 goes into 42 7 times, so n=7 the group size.
We need to pick 42 out of 45 numbers and there are (45)(44)(43)/6=14,190 ways of doing this.
Now we come to the question of how many different groups each containing 7 members.
So we have 42 numbers and we can select any 6. There are 5,245,786 different selections. Then we're left with 36 numbers, and there are 1,947,792 different selections for the second group. We're left with 30 numbers, and 593,775 different selections. Then 24 numbers and 134,596 selections, 18 numbers and 18,564 selections, 12 numbers and 924 selections, and the remaining 6 numbers left for the 7th group. The total is the product of all these, a very large number=
14,190×5,245,786×1,947,792×593,775×134,596×18,564×924×1.
But we have to dismiss the same combination of groups. For example, if the groups are labelled A-G, and let's say A is 1-6, B=7-12, and so on is one arrangement. Then another arrangement could include B with 1-6 and A with 7-12. We count them as being the same combination. In 7 groups there are 5,040 different arrangements, so we divide the above product by 5,040=1.43462×1033, if I'm right. Too many to list!
