Question

Harold has a collection of 20 movies that he wants to organize.

a.There is a set of small shelves that can store media at the foot of Harold’s bed. If each shelf can store up to 5 movies, how many different ways could movies be lined up on a shelf (if the shelf is going to be full)?

b.A friend of Harold’s asked to borrow some movies. How many different ways can Harold lend his friend 3 movies?

c.Unknown to Harold, his friend does not own a Blu-ray player. If 6 of Harold’s movies are stored on Blu-ray, what is the probability that at least one of the 3 movies Harold lends to his friend is on Blu-ray?

Answer 1

a) The shelves can store 5 movies each.

1st 5 movies can get selected from 20 movies in 20C5 = 15504 ways and for each such cases selected 5 movies can be arranged among themselves in 5! ways i.e. they can be lined up in 5! different ways => 1st shelf can be filled in 15504*5! = 1860480 ways.

then, another 5 movies can get selected from rest 15 movies in 15C5 = 3003 ways and for each such cases selected 5 movies can be arranged among themselves in 5! ways, => 2nd shelf can be filled in 3003*5! = 360360 ways

then, another 5 movies can get selected from rest 10 movies in 10C5 = 252 ways and for each such cases selected 5 movies can be arranged among themselves in 5! ways, => 3rd shelf can be filled in 252*5! = 30240 ways

then, another 5 movies can get selected from rest 5 movies in 5C5 = 1 way and 5 movies can be arranged among themselves in 5! ways, => 4th shelf can be filled in 120 ways

b) There are 20C3 = 1140 different ways Harold can lend his friend 3 movies.

c)6 out of 20 of Harold’s movies are stored on Blu-ray. => P(any of Harold’s movies are stored on Blu-ray) = 6/20 = 0.3

P(at least 1 of the 3 movies Harold lends to his friend is on Blu-ray)

= 1 - P(none of the 3 movies Harold lends to his friend is on Blu-ray)

= 0.675