Need to know how to calculate the number of distinct combinations

So I have three people.

Person 1.

Age group 1 thru 8
Health group 1 to 5

So there are 40 unique possibilities.. easy, right.. 8*5

Move to 2 people and I come up with 540 unique combinations. I know this because the sample size is small enough so I could do it manually..

Person 1

Age group 1-8
Halth group 1-5

Person 2

Age group 1-8
Health group 1-5

As a point of clarity..

Person 1 being a 1-1, person 2 being a 2-1 is considered the same thing as
Person 1 being a 2-1 and person 2 being a 1-1

Meaning order is not important. I sure hope I am explaining this well.

So now I want to add a third person.

Person 3

Age group 1-2
Health group 1-5


1-1, 1-2, 1-1 is considered a duplicate of
1-1, 1-1, 1-2 and is also a duplicate of
1-2, 1-1, 1-1

If this is making sense... I am trying to figure out the number of unique combinations for this three person mixture.