# Math Help - 2 doubts ....help!

1. ## 2 doubts ....help!

1. if $a^2+b^2 = c^2$ ,please show why $(a+b)$ and $c$ have the same parity?

2. proof this using a combinatorial argument:
$\sum\limits_{r=0}^{n}C_r^n 2^{r} = 3^{n}$

2. 2. Hint: $\displaystyle \sum_{r=0}^{n}\binom{n}{r}2^r = \sum_{r=0}^{n}\binom{n}{r}(1)^{n-r}2^r$.

3. 1) $(a+ b)^2= a^2+ 2ab+ b^2$
"2ab" is clearly even so the parity of $(a+ b)^2$ is the same as the partity of $c^2= a^2+ b^2$.

And, of course, since $(2n)^2= 4n^2$ and $(2n+ 1)^2= 4n^2+ 4n+ 1= 2(2n^2+ 2n)+ 1$ the parity of $a^2$ is the same as the parity of a.

4. @TheCoffeeMachine
yes thanks!
but i want a proof by a combinatorial argument.

5. Well, you can prove the binomial theorem (which is what TheCoffeeMachine used) with a combinatorial argument.