summation of x * (y choose x) binomial coefficients What does this summation simplify to?
$$ \sum_{x=0}^{y} \frac{x}{x!(y-x)!} $$
I was able to realize that it is equivalent to the summation of $x\dbinom{y}{x}$ if you divide and multiply by $y!$, but I am unsure of how to further simplify.
Thanks for the help!

If you toss a coin $y$ times, the probability of getting a head exactly $x$ times is $\dbinom yx 2^{-y}$.  And the expected number of times you get a head is obviously $y/2$.  But the expected number of times you get a head is also
$$
\sum_{x=0}^y x\binom yx 2^{-y}.
$$
So
$$
\sum_{x=0}^y x\binom yx 2^{-y} = \frac y2.
$$
Now multiply both sides by $2^y$.
