Answer
$P(I~like~all~4~songs)=\frac{1}{143}\approx0.06993$
Work Step by Step
Combinations of 13 distinct songs (I like or do not like) taken 4 at a time (the order in which the songs are selected does not matter):
$N(S)=~_{13}C_4=\frac{13!}{4!(13-4)!}=\frac{13!}{4!\times9!}=\frac{13\times12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1}{4\times3\times2\times1\times9\times8\times7\times6\times5\times4\times3\times2\times1}=715$
Combinations of 5 distinct songs I like taken 4 at a time (the order in which the songs are selected does not matter):
$N(4~songs~among~the~5~I~like)=~_{5}C_4=\frac{5!}{4!(5-4)!}=\frac{5!}{4!\times1!}=\frac{5\times4\times3\times2\times1}{4\times3\times2\times1\times1}=5$
Using the Classical Method (page 259):
$P(I~like~all~4~songs)=\frac{N(4~songs~among~the~5~I~like)}{N(S)}=\frac{5}{715}=\frac{1}{143}\approx0.06993$
