Solution: 2013 Fall Midterm - 2

Author: Pat Morin

Question

You are given 5 books and 7 bookshelves. How many ways are there to place these books on the shelves? (The order on the shelves matters.)

(a)

$\frac{11!}{7!}$

(b)

$\frac{12!}{7!}$

(c)

${7}\choose{5}$

(d)

$\frac{11!}{6!}$

The first book can be placed on any of the 7 shelves.
The second book can be placed in 8 spots because it can be put to the left of the first book or the right of the first book
The third book can be placed in 9 spots because it can be put to the left of the first book, between the first and second book, or to the right of the second book
The fourth book can be placed in 10 spots because it can be put to the left of the first book, between the first and second book, between the second and third book, or to the right of the third book
The fifth book can be placed in 11 spots because it can be put to the left of the first book, between the first and second book, between the second and third book, between the third and fourth book, or to the right of the fourth book

$ 7 \cdot 8 \cdot 9 \cdot 10 \cdot 11 $

$ = \frac{11!}{6!} $