Probability

Maths Help suggests:

In the British National Lottery, six ball are selected at random from
forty-nine numbered balls. Players have to guess which six balls will
be drawn. If they get all six correct, they win the jackpot prize.

We need to work out the total number of possible ways of choosing six
different numbers from forty-nine.
Note that the order of choosing the numbers does not matter.
For example, the numbers 2 , 7 , 34 , 21 , 46 , 11
give the same winning combination as 34 , 11 , 7 , 21 , 2 , 46.

• There are 49 possible choices for the first ball
• For each of these 49 choices of the first ball,
  there are 48 possible choices for the second ball
  (because one ball has already been taken out)
  meaning there are 49 x 48 ways of choosing the first two balls.
• However, because the order doesn't matter, this has to be divided by 2
  (because the second ball can come before or after the first ball)
  
  
• So the number of different possible choices for two balls can be written as
  
  
• There are then 47 possible choices for the third ball,
  but since it doesn't matter which position the third ball goes in
  
  we divide by 3, giving
  
  
• This sequence continues for all six balls which are chosen.

So the total number of possible ways of choosing six balls from forty-nine is

which is equal to
13 983 816
or approximately 14 000 000 (14 million).

Therefore, as you rightly said, the chance of a single ticket winning the
jackpot is approximately one-in-14million

USEFUL TIP: If your calculator has a button marked nCr
you can get the answer directly.
"n" stands for the total number (here n=49)
"r" stands for the number you want to select (here r=6)
So if you type in the sequence 49 nCr 6 =
you should get the result 13 983 816 directly.

nCr is shorthand for the number of different ways of choosing r items from n
where the order does not matter.