Question

A shelf contains 20 different books of which 4 are in single volume and the others form sets of 8, 5 and 3 volumes respectively. Number of ways in which the books may be arranged on the shelf, if the volumes of each set are together and in their due order?
(a) 20!
(b) 7!
(c) 8!
(d) 8 $\times$ 7!

Answer 1

The number of books is given to us. It is also given to us how of those are in single volume and how many of the others are in sets of volumes. We have to arrange the books on the shelf. It means they will be arranged linearly with a well defined starting and ending point. We will find how many entities there are which need to be arranged linearly. Then we find the number of ways that those can be arranged.

Complete step-by-step answer:
It is given to us that the shelf contains 20 books.
Number of books that are in single volumes are 4.
Therefore, these 4 will be separate entities.
Then, there is a set of 8 volumes. But since the books in the same volumes are to be kept together and are always arranged in their due order, this set of 8 books will be considered as the single entity.
Similarly, we have two more sets, one containing 5 volumes and the other containing 3 volumes. Same concept will be applied to these two sets as applied for the set of 8 volumes.
Now, the total entities will be equal to 4 separate books and 3 sets of volumes. Therefore, a total of 7 entities which are to be arranged in a linear fashion.
The first place from any end (say left) can be occupied in 7 ways, as there are 7 entities.
This leaves us with 6 entities and 6 places.
Next place can be occupied in 6 ways.
The place next to that in 5 ways.
We will fill the places in a similar manner until entities have a place.
Therefore, total number of ways the books can be arranged will be in $7\times 6\times 5\times 4\times 3\times 2\times 1=7!$

So, the correct answer is “Option (b)”.

Note: Students can directly remember that if there are n things which are to be arranged in n places in linear fashion, the number of ways it can be done is n!. If the arrangement is to be done in a circular manner, the number of ways will be (n – 1)!.