. Out of 10 white, 9 black and 7 red balls, in how many ways can we select one or more balls?

Answer 1

There are three ways to select one ball: R, B, W
There are 3 * 2 * 1 ways to select two balls: RR, RB, RW, BB, BW, WW
There are 4 * 3 * 2 * 1 ways to select four balls
There are 5 * 4 * 3 * 2 * 1 ways to select five balls

This is refered to as 5! ways... 5 * 4 * 3 * 2 * 1

This continues until you hit the number of ways to select 8 balls, because you can't have sevel red balls. You'd have 8! - 1
For 9 balls, you'd have 9! - 2! (there are 2 ways to have more than 7 red balls)
For 10 balls, you'd have 10! - 3! - 1 (3 ways to have more than 7 red balls, 1 way to have more than 9 black balls)

It proceeds like this.

The actual odds are calculated differently.