- #1
lveenis
- 13
- 0
Homework Statement
Hi everyone,
In my assignment I've been asked to show that 2^n = Ʃ(nCi) i from 0 -> n
ie: 2^n = nC0 + nC1 + ... + nCn and I have to do this by induction and then also by a combinatorial argument.
Homework Equations
Right now I'm just working on the induction part.
BASE CASE: n = 0
ASSUME: 2^n = ƩnCi
PROVE: 2^(n+1) = Ʃ(n+1)Ci
The Attempt at a Solution
I've shown the base case, 2^0 = 1 and 0C0 = 1
Then I assumed that 2^k = ƩkCi (i from 0->k)
Now I've expanded (n+1)Ci = (n+1)C0 + (n+1)C1 +... + (n+1)C(n+1)
I know there must be a way to do this simply using algebra but I'm totally stumped.
I tried saying that 2^k+1 = 2(2^k) which we know from the assumption to be 2*(ƩkCi) but that doesn't seem to help me out.
If anyone could give me a hint which direction to go with the algebra I would really appreciate it!