Answer
$P(I~like~3~of~them)=\frac{16}{143}\approx0.1119$
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 3 at a time (the order in which the songs are selected does not matter):
$N(3~songs~among~the~5~I~like)=~_{5}C_3=\frac{5!}{3!(5-3)!}=\frac{5!}{3!\times2!}=\frac{5\times4\times3\times2\times1}{3\times2\times1\times2\times1}=10$
Combinations of $13-5=8$ distinct songs I do not like taken 1 at a time (the order in which the songs are selected does not matter):
$N(1~song~among~the~8~I~do~not~like)=~_{8}C_1=\frac{8!}{1!(8-1)!}=\frac{8!}{1!\times7!}=\frac{8\times7\times6\times5\times4\times3\times2\times1}{1\times7\times6\times5\times4\times3\times2\times1}=8$
Using the Multiplication Rule of Counting (page 298):
$N(I~like~3~and~I~do~not~like~1)=10\times8=80$
Using the Classical Method (page 259):
$P(I~like~3~of~them)=\frac{N(I~like~3~and~I~do~not~like~1)}{N(S)}=\frac{80}{715}=\frac{16}{143}\approx0.1119$
