Math CentralQuandaries & Queries


Question from Mark, a teacher:

Can you explain why the lotto 6/49 (a lottery of 49 numbers and 6 slots) is a combination and not a permutation?

It seems to me that it would a permutation given that the numbers are written in numeric "order" when you look them up in the newspaper.

I realize that 49P6 is orders of magnitude larger than 49C6. But I am confused about the reasoning behind it.


In a permutation, order matters. If the numbers represented a permutation, you would have to specify an order when you bought your ticket and it would have to match the order in which the balls were drawn from the machine. If your numbers were 3 24 12 33 1 19 and the balls in order were 3 24 12 33 19 1 you would not win a jackpot (though presumably some smaller prize!) With those rules, the numbers would not be printed in numeric order, but in draw order; it's only because the numeric order is unimportant that they can sort them without losing essential information.

This would be a bad thing unless the lottery were being organized on a global scale (and with tickets at a price that most humans could afford) because only one ticket in about ten billion would win, and people would get tired of waiting (or start rumors that the lottery was rigged not to pay jackpots). Standard 6/49 rules give one chance in thirteen million, about the right scale for a lottery in a medium-sized country. (If there were too few possibilities, the jackpots would usually be divided many ways and shares might be disappointingly small.)

Good Hunting!

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