A 26 Volume Encyclopedia is Placed on a Bookshelf

A 26 Volume Encyclopedia is Placed on a Bookshelf.

An encyclopedia consists of 8 volumes, numbered i to 8, initially bundled in ascending order of their numbers.

a) How many means tin can nosotros put these volumes on a shelf?

You lot are correct that the viii volumes tin be permuted in
$8!$
means.

b) In how many ways can we put these volumes on a shelf so that at least one book is not occupying the aforementioned starting position?

At least ane volume does not occupy the same starting position unless all the volumes are placed in their original positions. The only permutation that leaves all volumes in their original position is the identity permutation
$(1, ii, 3, 4, v, 6, 7, 8) \to (1, ii, 3, 4, v, 6, 7, 8)$, so there are
$viii! – 1$
permutations that leave at least one book non in its starting position.

c) In how many means can we place these volumes on a shelf so that exactly one volume is not placed in ascending order? For example, 2, 1, three, 4, five, vi, 7, 8 or i, ii, 4, v, 3, 6, 7, 8.

There are eight means to select the volume that is out of identify and seven other positions to which that volume can exist moved. For instance, if we select
$3$
and motion information technology to the fifth position, we obtain the sequence
$(1, 2, 4, 5, three, 6, 7, eight)$
since the remaining seven numbers must be placed in ascending order. This suggests that there are
$8 \cdot 7$
sequences in which exactly one book does not appear in ascending society.

However, we have counted each instance in which we switched the positions of two adjacent numbers twice, once when nosotros moved the larger number to the smaller number’s position and once when we moved the smaller number to larger number’southward position. For instance, if we move
$1$
to the second position, nosotros obtain the sequence
$(2, 1, iii, 4, 5, 6, 7, viii)$. Nosotros as well obtain the sequence
$(2, 1, 3, iv, 5, 6, vii, 8)$
if we move
$2$
to the starting time position. There are seven pairs of next numbers.

Therefore, in that location are
$8 \cdot 7 – 7 = vii \cdot vii$
sequences in which exactly one volume does not announced in ascending order.

d) In how many ways can we select 3 of the 8 volumes and so that they practice not have consecutive numbers?

This is equivalent to request in how many ways we can line upwards five bluish and three greenish assurance and then that no two of the green assurance are consecutive. Line upwardly the five blue balls.

five_blue_balls_arranged_in_a_row

Doing and so creates six spaces in which to place the green assurance, four between successive blueish assurance and two at the ends of the row. To ensure that no two of the green assurance are consecutive, nosotros must choose three of these six spaces in which to identify 1 green brawl each, which can be done in
$\binom{6}{iii}$
ways. For example, if we select the third, fourth, and 6th spaces, we obtain the arrangement shown below.

five_blue_and_three_green_balls_arranged_in_a_row_so_that_no_two_green_balls_are_consecutive

Now number the balls from left to right. The numbers on the light-green balls are the numbers of the volumes on the selected encyclopedias, no two of which are consecutive.

A 26 Volume Encyclopedia is Placed on a Bookshelf

Source: https://math.stackexchange.com/questions/4261596/an-encyclopedia-consists-of-8-volumes-numbered-1-to-8-initially-arranged-in-as