Answer
$P(I~like~2~of~them)=\frac{56}{143}\approx0.3916$
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 2 at a time (the order in which the songs are selected does not matter):
$N(2~songs~among~the~5~I~like)=~_{5}C_2=\frac{5!}{2!(5-2)!}=\frac{5!}{2!\times3!}=\frac{5\times4\times3\times2\times1}{2\times1\times3\times2\times1}=10$
Combinations of $13-5=8$ distinct songs I do not like taken 2 at a time (the order in which the songs are selected does not matter):
$N(2~songs~among~the~8~I~do~not~like)=~_{8}C_2=\frac{8!}{2!(8-2)!}=\frac{8!}{2!\times6!}=\frac{8\times7\times6\times5\times4\times3\times2\times1}{2\times1\times6\times5\times4\times3\times2\times1}=28$
Using the Multiplication Rule of Counting (page 298):
$N(I~like~2~and~I~do~not~like~2)=10\times28=280$
Using the Classical Method (page 259):
$P(I~like~2~of~them)=\frac{N(I~like~2~and~I~do~not~like~2)}{N(S)}=\frac{280}{715}=\frac{56}{143}\approx0.3916$
