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:

(a) 3628800 ways

(b) 17280 ways

(c) 120960 ways

Step-by-step explanation:

Given

[tex]Novels = 5[/tex]

[tex]Maths = 4[/tex]

[tex]Biology = 1[/tex]

[tex]Total = 10[/tex]

Solving (a): Arrangement with no restriction.

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

So, the number of arrangement is:

[tex]Arrangement =10![/tex]

[tex]Arrangement =10*9*8*7*6*5*4*3*2*1[/tex]

[tex]Arrangement =3628800[/tex]

Solving (b): Maths book together and Novels together

First, arrange the 4 maths books as:

[tex]Maths = 4![/tex]

Next, arrange the 5 novels as:

[tex]Novels = 5![/tex]

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:

[tex]Books = 3![/tex]

Total arrangement is:

[tex]Total = 4! * 5! * 3![/tex]

[tex]Total = 4*3*2*1 * 5*4*3*2*1 * 3*2*1[/tex]

[tex]Total = 17280[/tex]

Solving (c): Maths book together

First, arrange the 4 maths books as:

[tex]Maths = 4![/tex]

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:

[tex]Books = 7![/tex]

Total arrangement is:

[tex]Total = 4! * 7![/tex]

[tex]Total = 4*3*2*1 * 7*6*5*4*3*2*1[/tex]

[tex]Total = 120960[/tex]

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