Binomial Coeffieients

Probability Level 4

$\large \displaystyle\sum_{i=0}^{264}(-1)^{i}\binom {529}{2i}$

If the value of the summation above is in the form $a^b$ , where $a$ and $b$ are positive integers with $a$ is minimized, enter $a+b$ .

Notation : $\binom MN$ denotes the binomial coefficient , $\binom MN = \frac{M!}{N!(M-N)!}$ .

Don't ask why 529.

The answer is 266.

1 solution

展豪張
Apr 29, 2016

Relevant wiki: Binomial Theorem

The expression
$=\Re((1+i)^{529})$
$=\Re((\sqrt 2 e^{i\frac \pi 4})^{529})$
$=\Re(2^{529/2}e^{i\frac \pi 4})$
$=2^{529/2}\Re(e^{i\frac \pi 4})$
$=2^{529/2}\Re(\cos\dfrac\pi 4+i\sin\dfrac\pi 4)$
$=2^{529/2}\dfrac 1{\sqrt 2}$
$=2^{528/2}$
$=2^{264}$
The answer $=2+264=266$

I solved it same way.

Hana Wehbi - 4 years, 9 months ago

Why 529? 😂😂.

Anurag Pandey - 4 years, 8 months ago

No...don't ask why......

展豪張 - 4 years, 8 months ago

Binomial Coeffieients

Don't ask why 529.

The answer is 266.

1 solution

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