Thread: Combinations giving me a headache

I've watched several videos and read several articles but nothing seems to help.

I can't for the life of me get a firm grasp of combinations. I have a very easy time realizing permutations. It's logical, I have 3 spots and 5 people to fill it. In the first spot I have 5, in the second I have 4 and the third I have 3. However, combinations just seem to complicate everything and it seems no one has a simple LOGICAL explanation for them.

Can anyone here please point me in the right direction? Thanks.

Hey Paze.

Basically combinations don't distinguish between the ordering.

As an example lets say we have a sequence of observations with either an A or a B. A few examples of length five include AABBA and ABABA.

Now in combinations, we don't care about the order of the sequence but only that the number of elements of a given type is the same.

In a permutation we distinguish ABABA from AABBA but in combinations we only care that we have three A's and two B's.

This is why in combinations we have a lower amount and this is compensated by a factorial.

nPr = n!/(n-r)! and nCr = n!/[r!*(n-r)!].

With the combinations, we are looking for the number of times we get a sequence with a certain number of events. A sequence that has two events is called a Binomial distribution and the distribution with n events is a Multinomial.

The above happens when every element of the sequence has the same probability as any other element.

So thats the main thing: in combinations we don't care about order but only that we have a sequence with a specific count of events (so in binomial, we have so many A's and so many B's but the order or arrangement is not important).

I understand the part where we are not ordering things, but how does that work with that formula? In essence we have 3 spots and 5 people to fill it...We don't care about the order so 3 spots are filled with 3 people and then we just arrange the last 2 people so it becomes 3+1+1 = 5 ways...?

Am I understanding it right? Guess not..The answer should read 10

Originally Posted by Paze

I understand the part where we are not ordering things, but how does that work with that formula? In essence we have 3 spots and 5 people to fill it...We don't care about the order so 3 spots are filled with 3 people and then we just arrange the last 2 people so it becomes 3+1+1 = 5 ways...? Am I understanding it right? Guess not..The answer should read 10

When you write "3 spots are filled with 3 people and then we just arrange the last 2 people so it becomes 3+1+1 = 5 ways...?",you are still thinking about order. Here is a different model.
Say we have seven people from which we pick three.
The string can be rearranged in ways. Right?

If we have an alphabetical listing of the seven people, the any of those strings selects three people:
the one that are assigned an X.

Combinations are about content.

Gereralize: Say we have N people from which we pick k.
The combination , the number of ways to arrange

Combinations are a grouping of items where order doesn't matter. Think of cards

a hand with 2 of clubs, 9 of hearts, and 8 of diamonds

is the same thing as

a hand with 9 of hearts, 8 of diamonds, 2 of clubs

The order of the cards in your hand does not matter.