Question

An English teacher needs to pick 9 books to put on his reading list for the next school year, and he needs to plan the order in which they should be read. He has narrowed down his choices to 13 novels, 17 plays, and 13 nonfiction books. If he wants to include an equal number of novels, plays, and nonfiction books, how many different reading schedules are possible

Answer 1

Different schedule Possible = 55,621,280

Explanation:

Given - An English teacher needs to pick 9 books to put on his reading list for the next school year, and he needs to plan the order in which they should be read. He has narrowed down his choices to 13 novels, 17 plays, and 13 nonfiction books.

To find - If he wants to include an equal number of novels, plays, and nonfiction books, how many different reading schedules are possible ?

Formula used -

Number of combinations of n things taking r at a time :
`^(n)C_(r)` =
`(n!)/(r! (n-r)!)`

Proof -

Given that, An English teacher needs to pick 9 books and wants to include equal number of novels, plays, and nonfiction books

i.e. he will Choose

3 books from Novels

3 books from Plays

3 books from Non-fiction books

Now,

Given that,

Total number of novel = 13

Total number of plays = 17

Total number of non-fiction books = 13

So,

Different schedule Possible =
`^(13)C_(3)* ^(17)C_(3)*^(13)C_(3)`

=
`(13!)/(3! (13-3)!)* (17!)/(3! (17-3)!)* (13!)/(3! (13-3)!)`

=
`(13!)/(3! (10)!)* (17!)/(3! (14)!)* (13!)/(3! (10)!)`

=
`(13(12)(11)10!)/(3! (10)!)* (17(16)(15)14!)/(3! (14)!)* (13(12)(11)10!)/(3! (10)!)`

=
`(13(12)(11))/(3(2)(1))* (17(16)(15))/(3(2)(1))* (13(12)(11))/(3(2)(1))`

=
`{13(2)(11)}*{17(8)(5)}* {13(2)(11)}`

= 55,621,280

⇒Different schedule Possible = 55,621,280