1. ## 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.

2. ## Re: Combinations giving me a headache

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).

3. ## Re: Combinations giving me a headache

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...?

4. ## Re: Combinations giving me a headache

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 $XXX0000$ can be rearranged in $\frac{7!}{(3!)(4!)}$ 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.

.

Gereralize: Say we have N people from which we pick k.
The combination $\binom{N}{k}=\frac{N!}{(k!)([N-k]!)}$, the number of ways to arrange $k\text{ X's and }(N-k)\text{ 0's .}$

5. ## Re: Combinations giving me a headache

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.