Distribute 5 red balls and 5 blue balls into three distinct boxes So if I was just doing them separately there would be 21 ways or 7 choose 5 ways. Do i just multiply it then 21 * 21? Unsure how to put them together.

You correctly calculated the number of ways to put 5 red indistinguishable balls into 3 distinguishable boxes.
Same for 5 blue balls.
There is an obvious 1-to-1 relation between "combination of 5 blue and 5 red balls" and a pair ("combination of red balls", "combination of blue balls"). So 21 * 21 is the answer.
