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Combinatorics confusion

Combinatorics confusion


While solving a problem how do we choose between variation or combination or permutation? like its always a difficult part for me to differentiate what a question is asking for.

1 Answer

365 Team

Hey Steven,
Thanks for reaching out!
So, I’m going to quickly give you some pointers when dealing with variations, combinations and permutations.
Let’s start with permutations since they’re the most “restrictive”. To use Permutations, we simply have “N”-many elements, which we need to arrange or split in some way. However, we are using all “N” elements here. For instance, if you go back to the Formula 1 race example, we use Permutations because every driver gets a position at the end of the race. (Even the ones that can’t finish get placed depending on how much of the race they finished.)
Okay, next, if we want to split or arrange some (but not necessarily all) elements, then we use variations. For instance, imagine you were judging a ballroom dancing competition where there were 10 teams competing. You want to know how many different ways the 3 medals (gold, silver and bronze) could be split up among the competitors. Since you don’t really care about who finishes 4th through 10th, it’s as if you’re only arranging 3 teams, but have 10 teams to choose from.
Lastly, we’re down to combinations. For them, we only need to “pick”, but not arrange certain elements. For instance, imagine you’re going on a short holiday and you need to pack your bags, but there’s only room for 4 of your favorite T-shirts. Now, (assuming) you don’t really care what order you place them in your suitcase, it’s only a matter of choosing the 4 T-shirts. Therefore, we use combinations.
Now, to briefly summarize these, here are the rules of thumb you should be following:
1). Arranging all the elements -> Permutations
2). Arranging some of the elements (a.k.a picking & arranging) -> Variations
3.) Picking some of the elements -> Combinations
4.) To “arrange” the elements, we need some sort of “order”. This “order” can be numeric (e.g. 1st, 2nd. 3rd) or non-numeric. In the latter, there is no fixed hierarchy, so we assign this order arbitrarily. To elaborate, we have different positions, and who/what goes where/when matters, but there is not set scheme put in place. Like, if you want to pain your house, every room is a different “position” and it matters which room you paint in which color. However, the rooms aren’t numbered (usually), so the “order” is arbitrarily set by us.
Hope this helps!
365 Team

Thanks for providing the rules of thumb. It helps to clarify when to use each concept.

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