Calculation explained in choosing 6 from 49

In a typical 6/49 game, six numbers are drawn from a range of 49 and if the six 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 probability of this happening is 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 first ball selected from the bag. Accordingly, there are 49 different ways of choosing that first number. When the draw comes to the second number, there are now only 48 balls left in the bag (because the balls already drawn are not returned 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 × 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 × 48 ways, so the chances of correctly predicting 3 numbers drawn from 49 is calculated as 49 × 48 × 47. This continues until the sixth number has been drawn, giving the final calculation, 49 × 48 × 47 × 46 × 45 × 44, which can also be written as . 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 is to understand that the order of the 6 numbers is not significant. That is, if a ticket has the numbers 1, 2, 3, 4, 5, and 6, it wins as long as all the numbers 1 through 6 are drawn, no matter what order they come out in. Accordingly, given any set of 6 numbers, there are 6 × 5 × 4 × 3 × 2 × 1 = 6! or 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

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In most popular spreadsheets, this function, called the combination function is denoted 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
Taken as a class, the number of possible combinations for a given lottery can be referred to as the "number