Prove Summation: 2n= \sumnk (^{n}_{k}) (^{m-n}_{n-k}) = (^{m}_{n})

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Homework Statement



prove that
\sumnk=0 (^{n}_{k}) (^{m-n}_{n-k}) = (^{m}_{n})

Homework Equations


2n=\sum^{n}_{k=0} (^{n}_{k})


The Attempt at a Solution


I know that every summand on the left hand side is a member of the power set times the combinations of it's complement and we can think of them in terms of the set changing with every possible combination, 1 element at a time and 2 at a time, 3 at a time, ..., n at a time. Or something like that... Definitely having trouble putting this into words let alone an equation. I certainly see the connection but am needing a little help on the proof. Please not the whole proof but just a little help putting it into words?
 
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Having some trouble getting my head around this still.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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