Combinatorics

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May 5th 2011, 12:23 PM
raed

Combinatorics

Dear Colleagues,

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

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

Quote:

Originally Posted by raed

Dear Colleagues,

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

Best Regards.

Hint:

Consider

$(1+1)^{20}$

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 $\binom {20}{0} +\binom {20}{2} +\binom {20}{4} +... \binom {20}{20}$ ?

$\left( {x + y} \right)^n = \sum\limits_{k = 0}^n {\binom{n}{k}x^k y^{n - k} }$ .
What if $x=1~\&~y=1~?$
May 5th 2011, 12:39 PM
raed

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

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 $\binom {20}{1}, \binom {20}{3},...,\binom {20}{19}$ ?

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

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