In a typical 6/49 lotto, 6 (k) numbers are drawn from a range of 49 (n) and if the 6 numbers on a ticket match the numbers drawn, the ticket holder is a jackpot winner – this is true no matter the order in which the numbers appear. The odds of this happening are 1 in 14 million (13,983,816 to be exact).
The relatively small chance of winning can be demonstrated as follows:
Starting with a bag of 49 differently-numbered lottery balls, there is clearly a 1 in 49 chance of predicting the number of the 1st ball out of the bag. Accordingly, there are 49 different ways of choosing that first number. When the draw comes to the 2nd number, there are now only 48 balls left in the bag (in case of no return of already drawn balls to the bag), so there is now a 1 in 48 chance of predicting this number.
Thus, each of the 49 ways of choosing the first number has 48 different ways of choosing the second. This means that the odds of correctly predicting 2 numbers drawn from 49 is calculated as: 49 x 48. On drawing the third number there are only 47 ways of choosing the number; but of course someone picking numbers would have gotten to this point in any of 49 x 48 ways, so the chances of correctly predicting 3 numbers drawn from 49 is calculated as: 49 x 48 x 47. This continues until the sixth number has been drawn, giving the final calculation: 49 x 48 x 47 x 46 x 45 x 44 (also written as 49! / (49-6)!). This works out to a very large number (10,068,347,520), which is however much bigger than the 14 million stated above.
The last step needed to understand that the order of the 6 numbers is not significant. That is, if a ticket has the numbers 01, 02, 03, 04, 05, 06 – it wins as long as all the numbers 1 through 6 are drawn, no matter what order they come out. Accordingly, given any set of 6 numbers, there are 6 x 5 x 4 x 3 x 2 x 1 = 6 factorial = 6! = 720 ways they could be drawn. Dividing 10,068,347,520 by 720 gives 13,983,816, also written as 49! / (6!·(49-6)!), or more generally as
C(n/k) = (n!)/(k!(n-k)!).
In most popular spreadsheets, the combinations function is COMBIN(n,k). For example, COMBIN(49,6) (the calculation shown above), would return 13,983,816. For the rest of this article, we will use the notation c(n,k) for convenience.
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