Problem 1:
If $n$ people are seated in a random manner in a row containing $2n$ seats, what is the probability that no two people will occupy adjacent seats?
I know that the probability that $n$ people are seated in $2n$ rows is ${2n\choose n}$
I also know that the answer to this problem is $n+1\over {2n\choose n}$
Why does $n+1$ solidify that the people will not be sitting next to each other? I somewhat understand it, but I would appreciate a more logical explanation.
Problem 2:
What is the probability that a group of $50$ senators selected at random will contain one senator from each state?
So obviously the set of $50$ senators is the combinatorial ${100\choose 50}$ and we want $1$ senator from each state so that's ${2\choose 1}$.
So we have ${2\choose 1}\over {100\choose 50}$, but the ${2\choose 1}$ should be ${2\choose 1}^{50}$ for all 50 states. Why do we have to put the numerator to the 50th power? Shouldn't ${2\choose 1}$ suffice that each senator should be from $1$ state?
Thanks