Question

Deidre is picking out some movies to rent, and she is primarily interested in documentaries and children's movies. She has narrowed down her selections to 17 documentaries and 20 children's movies. How many different combinations of 3 movies can she rent if she wants at least one documentary?

Answer 1

Answer: 6630

Step-by-step explanation:

Given , Number of documentaries = 17

Number of children's movies = 20

Total movies = 17+20=37

Number of combinations of r things taken out of things = $^nC_r=\dfrac{n!}{r!(n-r)!}$

Now, the number of different combinations of 3 movies can she rent if she wants at least one documentary

= (1 documentary+2 children's movies  , 2 documentary+1 children's movies , 3 documentary+0 children's movies)

$=^{17}C_1\times^{20}C_{2}+^{17}C_2\times^{20}C_{1}+^{17}C_3\times^{20}C_{0}$

$=(17)\times\dfrac{20!}{2!18!}+\dfrac{17!}{2!15!}\times(20)+\dfrac{17!}{3!14!}(1)\\\\=3230+2720+680=6630$

Hence, the  number of different combinations of 3 movies can she rent if she wants at least one documentary is 6630 .