Keno in Reno

I don't care for gambling that much, but when I do gamble I'm well aware of the odds of the games, and I certainly don't have any "lucky feelings" come over me when I'm trying to roll a hard six. Last month, we were in Reno sitting at a cafe, and Isa wanted to pick Keno numbers and I said yes, all the while explaining to her that Keno has some really dismal odds of winning. After each round of numbers was chosen, the Keno lady would come around the table, and I'd hand her a fresh sheet of numbers that Isa had chosen. She sighed and said, "You need to keep the same numbers! You will never win doing it this way!"
I smiled and didn't say anything and she left with our new numbers.
I looked over at Isa and asked,"Do you think if an 11 was chosen this time around, it has any bearing on whether or not an 11 is chosen the next game?"
"No," she replied.
"Do you think if we keep using the same numbers it increases our chances of winning?"
"I don't think it makes a difference," she said.
"Nice," I said, and then we promptly lost again. It's important to me that everyone knows the Keno lady is wrong.

Simplify This

    (a+x)(b-x)(c+x)(d-x) ... (z-x)

"Simplify" means that the solution can be written using fewer characters than the problem itself uses.

The Locker Puzzle

This is my favorite puzzle.

A hallway in a school has 100 lockers. A student runs down the hall and opens every locker. A second student runs down the hall and closes every other locker, starting at the second. A third student runs down the hall and "flips" every third locker, starting at the third - if the locker is open, he closes it; if it's closed, he opens it.

A total of 100 students run down the hall, opening and closing lockers. In general, the n-th student flips every n-th locker.

After all 100 students run through the hall, which lockers are open?