During the practical lecture in the probability course,
i am currently having issues understanding when we apply variations with and without repetitions and also when to apply combination with and without variation
Thanks for reaching out and sorry for the late response!
Let’s start with the difference between variations and combinations 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 repetition part it\’s usually pretty straightforward. If we have only 1 instance of something (be it a person, or an object), we can’t really repeat it. However, when have multiple (or infinite) of each “thing” we’re arranging, then we use repetition. For example – you’re making an ice-cream Sundae with 3 scoops of ice-cream. There is enough ice-cream available that you can have multiple scoops of the same one – hence repetition. Alternatively, we might be choosing a team of 3 people to send to a conference. We can’t send Jack twice (because there\’s only one of him), so we don\’t assume repetition.
Hope this helps!
Best, 365 Vik