# Combinatorics

• May 5th 2011, 12:23 PM
raed
Combinatorics
Dear Colleagues,

What is the value of $\displaystyle \binom {20}{0} +\binom {20}{2} +\binom {20}{4} +... \binom {20}{20}$ ?

Best Regards.
• May 5th 2011, 12:28 PM
TheEmptySet
Quote:

Originally Posted by raed
Dear Colleagues,

What is the value of $\displaystyle \binom {20}{0} +\binom {20}{2} +\binom {20}{4} +... \binom {20}{20}$ ?

Best Regards.

Hint:

Consider

$\displaystyle (1+1)^{20}$

Use the binomial theorem
• May 5th 2011, 12:32 PM
Plato
Quote:

Originally Posted by raed
Could you please help me in solving the following problem: What is the value of $\displaystyle \binom {20}{0} +\binom {20}{2} +\binom {20}{4} +... \binom {20}{20}$ ?

$\displaystyle \left( {x + y} \right)^n = \sum\limits_{k = 0}^n {\binom{n}{k}x^k y^{n - k} }$.
What if $\displaystyle x=1~\&~y=1~?$
• May 5th 2011, 12:39 PM
raed
Thank you very much for your reply. In fact I tried that but what about terms $\displaystyle \binom {20}{1}, \binom {20}{3},...,\binom {20}{19}$ ?

Best Regards.
• May 5th 2011, 12:43 PM
Plato
Quote:

Originally Posted by raed
Thank you very much for your reply. In fact I tried that but what about terms $\displaystyle \binom {20}{1}, \binom {20}{3},...,\binom {20}{19}$ ?

$\displaystyle x=1~\&~y=-1$ then what?