Question

Juan is picking out some movies to rent, and he is primarily interested in horror films and mysteries. He has narrowed down his selections to 77 horror films and 1313 mysteries. How many different combinations of 33 movies can he rent if he wants at least one horror film?

Answer 1

Answer:  854

Step-by-step explanation:

Given , Number of horror films = 7

Number of mysteries = 13

Total movies = 13+7=20

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

Now , the number of  combinations of 3 movies can he rent if he wants at least one horror film = $^7C_1\times^{13}C_2+^7C_2\times^{13}C_1+^7C_3\times^{13}C_0$

$=(7)\times\dfrac{13!}{2!(13-2)!}+\dfrac{7!}{2!(7-2)!}\times(13)+\dfrac{7!}{3!(7-3)!}(1)\\\\=546+273+35\\\\=854$

Hence, there are 854 different combinations of 3 movies can he rent if he wants at least one horror film .