Question

I have a specific question.
Can someone help clarify a solution for me (posted below)
Thanks

Answer 1

where does the
\[\left(\begin{matrix}100 \\ 50\end{matrix}\right)\]
derive from?

Answer 2

sorry i should give the context, the question was find the avg value for sin^100 x
i understand the process, but don't see how they obtain 100 choose 50 when the integral of every term is 0 for k not equal 0

Answer 3

Ok

Answer 4

the definition given of the integral of exp(ikx), says that if k is not equal to 0 then the integral is 0

Answer 5

we must integrate sin(x)^100, that is [(exp(ix) - exp(-ix))/2i]^100
right?

Answer 6

so...
sin(x)^100 = [(exp(ix) - exp(-ix))/2i]^100
= (1/2^100)*(exp(ix) - exp(-ix))^100

Answer 7

remember that i^100 = 1

Answer 8

now, for this part:
[(exp(ix) - exp(-ix))/2i]^100
use newton binomial formula

Answer 9

[(exp(ix) - exp(-ix))/2i]^100 =
\[\sum_{k=0}^{100} (-1)^{100-k} e^{ikx} e^{-i(100-k)x}\]

Answer 10

grouping these exponentials in one, we get:
\[\sum_{k=0}^{100} (-1)^{100-k} e^{i(2k+100)x}\]

Answer 11

now check when 2k+100 is not equal to 0

Answer 12

\[k \neq 50\]

Answer 13

so... only when k = 50, the integral is equal to 2*pi, at that time, from the newton binomial formula is at \[\left(\begin{matrix}100 \\ 50\end{matrix}\right)\]

Answer 14

so if we plug it all together, we find the result shown in the pic

Answer 15

got it?

Answer 16

yep got it, didn't think to combine the terms
thanks a lot