Answer :
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 = [tex]^nC_r=\dfrac{n!}{r!(n-r)!}[/tex]
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)
[tex]=^{17}C_1\times^{20}C_{2}+^{17}C_2\times^{20}C_{1}+^{17}C_3\times^{20}C_{0}[/tex]
[tex]=(17)\times\dfrac{20!}{2!18!}+\dfrac{17!}{2!15!}\times(20)+\dfrac{17!}{3!14!}(1)\\\\=3230+2720+680=6630[/tex]
Hence, the number of different combinations of 3 movies can she rent if she wants at least one documentary is 6630 .