Question

In how many ways can 5 different novels, 4 different mathematics books, and 1 biology book be arranged on a bookshelf if: (a) The books can be arranged in any order? Your answer is: 3628800 (b) The mathematics books must be together and the novels must be together? Your answer is : 6 (c) The mathematics books must be together but the other books can be arranged in any order? Your answer is: 5040

Answer 1

(a) 3628800 ways

(b) 17280 ways

(c) 120960 ways

Explanation:

Given

`Novels = 5`

`Maths = 4`

`Biology = 1`

`Total = 10`

Solving (a): Arrangement with no restriction.

We simply count each book with no restriction. i.e. 10 books

So, the number of arrangement is:

`Arrangement =10!`

`Arrangement =10*9*8*7*6*5*4*3*2*1`

`Arrangement =3628800`

Solving (b): Maths book together and Novels together

First, arrange the 4 maths books as:

`Maths = 4!`

Next, arrange the 5 novels as:

`Novels = 5!`

Lastly, take the 4 maths book as [1], the 5 novels as [1] and the remaining [1] biology book.

So, we have: 3 books

Arrange 3 books, we have:

`Books = 3!`

Total arrangement is:

`Total = 4! * 5! * 3!`

`Total = 4*3*2*1 * 5*4*3*2*1 * 3*2*1`

`Total = 17280`

Solving (c): Maths book together

First, arrange the 4 maths books as:

`Maths = 4!`

Next, take the 4 maths book as [1], then the remaining 6 books (i.e. 5 novels and 1 biology)

So, we have: 7 books

Arrange 7 books, we have:

`Books = 7!`

Total arrangement is:

`Total = 4! * 7!`

`Total = 4*3*2*1 * 7*6*5*4*3*2*1`

`Total = 120960`