I was running a training session for new maths teachers last week and over lunch someone mentioned the weekend’s huge lottery draw, with the largest jackpot in UK lottery history of £66m, shared by two winners.  I remember when the National Lottery started, one of my first thoughts about it being, “if someone had enough money they could buy up every combination and guarantee the jackpot”.  We talked about this over our jacket potatoes with chilli and I started to consider the practicalities.  And, of course, the maths.  So here are my musings…

Let’s begin with the old system.  The lottery numbers went from 1 to 49, from which you chose 6 distinct numbers and paid £1.  The number of possible combinations of six numbers out of 49 is 13,983,816.  This is calculated using the rather nice formula:

$\frac{49!}{6!43!}$

where the ! is read “factorial” and 49! = 49×48×47×…×1 or multiply 49 by every other integer down to 1.  It’s surprisingly hard to find a good explanation of why this is the formula, I might do one myself one day, but not now.

The theory goes that you could buy 13,983,816 lottery tickets and be a guaranteed winner, so as long as the jackpot was more than £14m you’d make a profit.  How long would it take you to buy that many tickets?  Assuming you had a really sympathetic newsagent and very strong muscles in your hand you might be able to fill out one set of numbers and put it through the machine every 10 seconds.  That’s 139,838,160 seconds of ticket buying, or just short of 1618½ days.