According to Powerball.com, the game is played like this
...we draw five white balls out of a drum with 69 balls and one red ball out of a drum with 26 red balls
Their odds explain that the change of you winning the jacketpot and matching the first five numbers plus the red ball are: 1 in 292,201,338. Why?
Why wouldn't it be (69*68*67*66*65)*26 = 1 in 35,064,160,560?
I have seen online some people use the COMBIN function in Excel to calculate this but it doesn't make sense to me?
Thank you!
On
When you calculates your answer of 35 billion you counted that there are 35 billion ways to choose a ticket where ORDER MATTERS. What this means is that we have counted the tickets $$1 2 3 4 5 r10$$ and $$5 4 3 2 1 r10$$ as different tickets, but since order doesn't matter with white balls, they aren't actually different. This we need to change how we count.
In this case, we really need to focus on how to count the number of ways to choose the 5 white balls where order doesn't matter. To do this we can use ideas of a combination.
The number $\binom{n}k$ is the number of ways that $k$ objects can be chosen from $n$ objects where order doesn't matter and repetition is not allowed. This is also sometimes denoted $C(n,k)$ or $nCk$. To calculate this, $\binom{n}k=\frac{n!}{(n-k)!k!}$.
With this in mind, let's see if we can't rework our ideas about the count. We are choosing 5 white balls from 69 so we get that there are $$\binom{69}5=\frac{69!}{64!\cdot5!}=11238513$$ to choose the white balls, then 26 ways to choose the red ball so we get $11238513*26=292,201,338$ ways to choose a ticket where order doesn't matter for the white balls.
If you have any questions about this feel free to ask for clarification.
Probably the order of the five white balls does not matter, so you have to divide by $5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ to account for the different orders in which these could be drawn.
Your calculation would be correct if the order did matter, i.e. if 1,2,3,4,5,6 wins, but not 2,1,3,4,5,6.