# Math Help - Binomial Thm

1. ## Binomial Thm

1.) Using the binomial theorem, prove:

(Not sure how to do chooose in maple, so ignore the - (it's not a fraction))

$\left(\frac{n}{0}\right) + \left(\frac{n}{1}\right)\cdot 2 + \left(\frac{n}{2}\right)\cdot 2^2 + \cdots + \left(\frac{n}{n}\right)\cdot 2^n = 3^n \, \, \forall$ integers $n \geq 0$

2.) Now prove the above giving a combinotrial proof.

2. $(1+2)^n = 3^n$

3. $3^n=(1+2)^n=\sum_{i=0}^{n}\binom{n}{i}(2^{i})(1)^{ n-i}=\sum_{i=0}^{n}\binom{n}{i}(2^{i})$