let 1<=r<=n. A subset of{1,2....n} of cardinality "r" is chosen at random. How do i calculate the probability that 1 is an element of the chosen subset
The number of subset of cardinality r is
$\displaystyle {{n} \choose {r}}$
and the number of those that doesn't contain 1 is $\displaystyle {{n-1} \choose {r-1}}$ as Plato told you.
The probability of taking a set that doesn't contain 1 is $\displaystyle {{n-1} \choose {r-1}}/{{n} \choose {r}}$