what is the difference between combination with repetition and variation with repetition?

Hey Luis,
Here is a response to a similar question explaining the difference between combinations and variations (w/o repetition):

Let's start with the difference between the two and work through a simple example.

So, the main distinction between the two is that combinations don't care about order, while variations do.

For instance, suppose you love tennis and you're a big fan of Djokovic, Nadal and Federer. You know all 3 men were in the tournament and 2 of them reached the final. If you simply care which 2 made the final, but not who won, we would use combinations because order does not matter.

Hence, if you only care about the match up, but don't care who actually ends up as the victor, you use combinatorics -> C(3,2) = 3!/(2!*1!) = 3. The 3 combinations are, obviously, Djokovic vs Nadal, Nadel vs Federer or Djokovic vs Federer.

Now, if we care who lifts the trophy, we use variations because order is relevant. Then, we have P(3,2) = 3! / 1! = 6 ways they 3 competitors can arrange.

1) Djokovic  2) Nadal 3) Federer,
1) Nadal 2) Djokovic, 3) Federer
1) Djokovic  2) Federer  3) Nadal,
1) Nadal 2)  Federer 3) Djokovic,
1) Federer 2) Nadal 3) Djokovic,
1) Federer 2) Djokovic, 3) Nadal

Thus, when some (or all) position matter, we are dealing with variations. For example, when we have to match banners to social media platforms in question 2, we have this artificial "order" because every position (platform) is different. The same distinction can be assigned to the tennis example, where we can name the positions: "winner", "runner-up" and "not in final". Essentially, as long as it matters who we put where, we have variations.

Now, as for the difference between the two when it comes to repetition, think of it this way - does it make a difference when we pick what? If no - we're using combinations, if yes - we're using variations. Once again, this all comes down to this artificial order, where the different positions matter. Hence, if we have to select what 3 flavors of ice cream to get for our Sundae, we'd use combinations with repetition, but if we're talking about different ways to repaint the house, then we're using variations (because it matters what color each room is).

Hope this helps out and don't hesitate to reach out if it doesn't.
Best,
365 Vik