- #1
noblerare
- 50
- 0
Homework Statement
This is actually part of a larger problem that asks us to prove that the number of ways of counting something is equal to [tex]3^n[/tex]. I have worked it out and the equation I get is:
[tex]\binom{n}{0}2^n + \binom{n}{1}2^{n-1}+\ldots+\binom{n}{n}2^{n-n}[/tex]
I am wondering how I should simplify this to make it equal to [tex]3^n[/tex]
2. The attempt at a solution
I rewrote the above equation into:
[tex]\displaystyle\sum_{i=0}^{n}\binom{n}{i}2^{n-i}[/tex]
But then I didn't know how to proceed from here since both the combinatorial choosing term and the powered terms are changing. I also tried factoring out [tex]2^n[/tex] but that didn't do anything.
Can anyone help me?
Thanks.