Q:

Number of groups of five balls such that the sum of all the balls is even

An urn contains 10 white balls, numbered 1-10, and ten black balls, numbered 1-10. A sample of five balls is chosen.
How many samples have the property that the sum of all of the balls is even?

I'm new to counting and not entirely sure how to do this. I was thinking of it as though the first four of the sample of five could be anything, then if the sum is even, you would have to pick another even number for the last ball, and if it's odd, you would have to pick an odd ball. but I don't know how many odd or even balls are left to pick from, unless I state whether the prevous selections were odd or even. Do I need to break it up into multiple cases, or is there an easier way?

A:

HINT: The sum is even if and only if the number of balls with odd numbers on them is even. Thus, you’ll get an even sum if and only if you have $0,2$, or $4$ balls with odd numbers. There are $10$ even-numbered balls and $10$ odd-numbered balls.

How many ways are there to choose $5$ even-numbered balls?  
How many ways are there to choose $3$ even-numbered balls and $2$ odd-numbered balls?  
How many ways are there to choose $1$ even-numbered ball and $4$ odd-numbered balls?  

