Puzzling
Favorite Answer
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.