Answer
$P(2~contain~diet~soda)=\frac{27}{220}\approx0.1227$
Work Step by Step
Combinations of 12 distinct cans (regular or diet soda) taken 3 at a time (the order in which the cans are selected does not matter):
$N(S)=~_{12}C_3=\frac{12!}{3!(12-3)!}=\frac{12!}{3!\times9!}=\frac{12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1}{3\times2\times1\times9\times8\times7\times6\times5\times4\times3\times2\times1}=220$
Combinations of 3 distinct diet soda taken 2 at a time (the order in which the cans are selected does not matter):
$N(2~diet~soda~among~the~3~diet~soda)=~_{3}C_2=\frac{3!}{2!(3-2)!}=\frac{3!}{2!\times1!}=\frac{3\times2\times1}{2\times1\times1}=3$
Combinations of 9 distinct regular soda I do not like taken 1 at a time (the order in which the cans are selected does not matter):
$N(1~regular~soda~among~the~9~regular~soda)=~_{9}C_1=\frac{9!}{1!(9-1)!}=\frac{9!}{1!\times8!}=\frac{9\times8\times7\times6\times5\times4\times3\times2\times1}{1\times8\times7\times6\times5\times4\times3\times2\times1}=9$
Using the Multiplication Rule of Counting (page 298):
$N(2~diet~soda~and~1~regular~soda)=3\times9=27$
Using the Classical Method (page 259):
$P(2~contain~diet~soda)=\frac{N(2~diet~soda~and~1~regular~soda)}{N(S)}=\frac{27}{220}\approx0.1227$
