Question

In: Statistics and Probability

A bookshelf contains 6 German books, 4 Spanish books and 7 French books. Each book is...

A bookshelf contains 6 German books, 4 Spanish books and 7 French books. Each book is different from one another. How many different arrangements can be done of these books if
(a) we put no restrictions on how they can be arranged?
(b) books of each language must be next to each other?
(c) all the French books must be next to each other?
(d) no two French books must be next to each other?

Expert Solution

6 German books, 4 Spanish books and 7 French books

How many different arrangements can be done of these books if

(a) we put no restrictions on how they can be arranged?

total number of books = 6 + 4 + 7 = 17

Number of ways all books can be arranged = 17!

( any book can be kept anywhere )

Number of ways all books can be arranged = 355687428096000

(b) books of each language must be next to each other?

Number of ways 6 German books can be arranged = 6!

Number of ways 6 German books can be arranged = 720

Number of ways 4 Spanish books can be arranged = 4!

Number of ways 4 Spanish books can be arranged = 24

Number of ways 7 French books can be arranged = 7!

Number of ways 7 French books can be arranged = 5040

Number of ways these three set of books can be arranged = 3!

Number of ways these three set of books can be arranged = 6

Total number of arrangements = 720*24*5040*6

Total number of arrangements = 522547200

(c) all the French books must be next to each other?

Number of ways 7 French books can be arranged = 7!

Number of ways 7 French books can be arranged = 5040

Remaining books = 17 - 7 = 10

Now, assuming French set as 1 book

Total number of arrangements = 5040*11!

Total number of arrangements = 39916800*5040

Total number of arrangements = 201180672000

(d) no two French books must be next to each other?

Total number of arrangements where no two French books must be next to each other = Number of ways all books can be arranged - Total number of arrangements all the French books must be next to each other 355687428096000 - 201180672000

Total number of arrangements where no two French books must be next to each other = 355486247424000

orchestra answered 1 year ago

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