a) ways to choose first five integers out of 59
$\binom{59}5$
+
ways to choose 6th integer
39
probability:$1\div( \binom{59}5 * 39 )=1\div195249054$
b)to choose 6th integer wrong :38ways (leave the right one)
probability:$38\div( \binom{59}5 * 39 )=38\div195249054=1\div5138133$
c) choose 3 correct and two wrong from first set : $\binom{5}{3}\binom{54}{2}$
and choose sixth correctly in 1 way
or
choose first four correct and one wrong :$\binom{5}{4}\binom{54}{1}$
and choose sixth wrong:38 ways
probability :$\frac{\binom{5}{3}\binom{54}{2}}{\binom{59}{5}}* \frac{1}{39} +\frac{\binom{5}{4}\binom{54}{1}}{\binom{59}{5}}*\frac{38}{39} = \frac{45}{357599}$
d)In this one i can't get book's answer as I've pulled an all nighter,may be later.