5. Suppose a small cinema with one screen has 5 screenings per day. The film dis

Question

5. Suppose a small cinema with one screen has 5 screenings per day. The film distributor has offered them 7 different films that they might screen (we will assume that none are particularly long, so the choice of films to show does not affect the available number of screenings Given that we are only interested in which films they are going to screen, not the order in which they show them, how many possible choices are there (a) If they choose to show all different films? (b) If three of the films are classified as "blockbusters" and they want to choose at least two blockbusters to be shown? (c) If they allow the possibility of showing the same film multiple times, and still want at least two screenings to be blockbusters (not necessarily different)? (d) If films can be repeated, they need at least two blockbuster screen- ings, but no film is to be shown more than 3 times throughout the day?

Explanation / Answer

(a)

Since order is not important, so the number of ways of choosing 5 different films out of a total of 7 films = 7C5 = 21

(b)

We have 3 blockbuster and 4 non-blockbuster movies.

Number of ways of choosing atleast 2 blockbuster movies = Choosing 2 blockbuster and 3 non-blockbuster movies + Choosing 3 blockbuster and 2 non-blockbuster movies = 3C2*4C3 + 3C3*4C2 = 18

(c)

In this case repetition of movies is allowed.

We have 3 blockbuster and 4 non-blockbuster movies.

Number of ways of showing atleast 2 blockbuster movies = Showing 2 blockbuster and 3 non-blockbuster movies + Showing 3 blockbuster and 2 non-blockbuster movies = (3^2)*(4^3) + (3^3)*(4^2) = 1008

(d)

The answer is same as calculated above, equal to 1008.

This is because in the above case, none of the film gets a chance to be shown more than 3 times. This is because since we want atleast 2 blockbuster movies and total allowed screenings are equal to 5,so the maximum number of times any movie can be shown is three, which occurs when we show the same blockbuster movie thrice, or the same non-blockbuster movie thrice.

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