what is the relationship between Pascal's Triangle and combinations?

Each item in Pascal's triangle relates to a combination.

First, remember that nCk (or sometimes it is written C(n,k)) is the number of ways when you have n items of choosing k of them.

nCk = C(n,k) = n! / (n-k)! k!

For example, if you had 1 item, there are C(1,0) ways of choosing none of them.

C(1,0) = 1 way.

Similarly:

C(1,1) = 1 way.

1 1

Well, that's not too exciting, but it continues from there.

C(2,0) = ways to pick 0 items out of 2 = 1 way

C(2,1) = ways to pick 1 item out of 2 = 2 ways

C(2,2) = ways to pick 2 items outof 2 = 1 way

1 2 1

Do you notice anything?

Let's continue then:

C(3,0) = 1 way

C(3,1) = 3 ways

C(3,2) = 3 ways

C(3,3) = 1 way

1 3 3 1

If you haven't noticed, this exactly matches Pascal's triangle. Each row represents the combinations of choosing 0 items, 1 item, 2 items, etc. out of n items.

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

etc.