Algebra 2: Combinations

There are 10 sophomores and 12 junior available tot form a committee. How many committees can be formed if 6 sophomores and 4 juniors must be on the committee?
10C6 and 12C4. but would you add them or multiply then and why?

multiply them, b/c it says 6 sophomores AND 4 juniors - they both have to happen. if you just added, it would have combinations that could have 4 juniors but no sophomores, etc. and wouldn't include all the possibilities (same combination of juniors w/ different combinations of sophomores, etc.)

We need 6 sophomores from a group of 10 so 10C6

We need 4 juniors from a group of 12 so 12C4

Now for the first combination of juniors we can allow every possible combinations of sophomores

For the second combination of juniors we can allow every possible combination of sophomores

etc.

So to get the total combination we we would keep adding together 10C6 12C4 times or 10C6 * 12C4

Each of the combinations of sophomores or juniors can be matched with all of the others, so you multiply.

im lost with the 10c6 and the 12c4 but only one committee can be formed cuz you would need two more sophomores to form a second committee
im rusty on my algebra