A fair coin is tossed $n$ times by two people. What is the probability that they get same number of heads? Say we have Tom and John, each tosses a fair coin $n$ times. What is the probability that they get same number of heads? 


I tried to do it this way: individually, the probability of getting $k$ heads for each is equal to $$\binom{n}{k} \Big(\frac12\Big)^n.$$ So, we can do $$\sum^{n}_{k=0} \left( \binom{n}{k} \Big(\frac12\Big)^n \cdot \binom{n}{k}\Big(\frac12\Big)^n \right)$$ which results into something very ugly. This ugly thing is equal to the 'simple' answer in the back of the book: $\binom{2n}{n}\left(\frac12\right)^{2n},$ but the equality was verified by WolframAlpha -- it's not obvious when you look at it. So I think there's a much easier way to solve this, can someone point it out? Thanks.
The probability John gets $k$ heads is the same as the probability John gets $n-k$ heads since the coin is fair.

So the answer to the original question is equal to the probability that the sum of Tom's and John's heads is $n$.  

That is the probability of $n$ heads from $2n$ tosses which is indeed $\frac{1}{2^{2n}}{2n \choose n}$.
The problem with this answer is that its not correct to begin with. Assume the problem pertains to two individuals throwing 2 coins then

the total outcomes is indeed $ \frac{1}{2^m} , m= 2n$ however the amount of possible outcomes is not $\binom{2n}{n}$ so in the case of each having 2 coins the resulting outcomes is:

1 head and two tails, this can be done 2 ways $2*2 = 4$

2 heads , this can be done only 1 way $1*1=1$

This is a total of $5$ and $\binom{2*2}{2} = 6$ this is of course not equal and difference of 1 holds as you continue.