How to calculate the probability of winning top prize of lottery?

[kenny1999]

Just take a simple example, Among 1-36 numbers, 6 balls are randomly drawn without putting back the ball. Only if you match the six numbers exactly you win the top prize, order doesn't matter.

I think it should be 1/36 * 1/35 * 1/34 * 1/33 * 1/32 * 1/31, is it?

[Moderator's note: moved from a technical forum.]

 

[FactChecker]

You have not accounted for "order doesn't matter". In your calculation, the order does matter. There are 6*5*4*3*2*1 = 720 ways to reorder what was drawn. To factor that in, multiply your answer by 720.
Another way to look at it is this:
The first draw would be a success if you got any of the 6 numbers: 6/36.
The second draw would be a success if you got any of the remaining 5 numbers: 5/35.
.
.
.
The last draw would be a success if you got the one remaining number: 1/31.
That gives the answer: 6/36 * 5/35 * 4/34 * 3/33 * 2/32 * 1/31 = 5.134$*10^{-7}$