Answer
See below
Work Step by Step
Substitute $n=50, r=4$ into the formula: $_n C_r=\frac{n!}{(n-r)!r!}$ to find a number of ways of choosing 4 movies out of your collection: $_{50} C_4=\frac{50!}{(50-4)!4!}=\frac{50!}{46!4!}=230,300$
If you pick two comedies, one drama and one thriller: $_{25} C_2 . _{15}C_1 . _{10} C_1=\frac{25!}{(25-2)!2!}.\frac{15!}{(15-1)!1!}.\frac{10!}{(10-1)!1!}=\frac{25!}{23!2!}.\frac{15!}{14!1!}.\frac{10!}{9!1!}=45,000$
If you pick one comedy, two dramas and one thriller: $_{25} C_1 . _{15}C_2 . _{10} C_1=\frac{25!}{(25-1)!1!}.\frac{15!}{(15-2)!2!}.\frac{10!}{(10-1)!1!}=\frac{25!}{24!1!}.\frac{15!}{13!2!}.\frac{10!}{9!1!}=26,250$
If you pick two comedies, one drama and two thrillers: $_{25} C_1 . _{15}C_1 . _{10} C_2=\frac{25!}{(25-1)!1!}.\frac{15!}{(15-1)!1!}.\frac{10!}{(10-2)!2!}=\frac{25!}{24!`!}.\frac{15!}{14!1!}.\frac{10!}{8!2!}=16,875$
There are $45,000+26,250+16,875=88,125$ ways.
The probability that you pick at least one DVD of each type of movie is $P(A)=\frac{88.125}{230.300}=0.383$
