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2. In how many ways can 3 different novels, 2 different mathematics books and 5 different chemistry books be arranged on a bookshelf if (Show your steps in this question) (a) (b) (c ) the books can be arranged in any order. (2 marks) the mathematics books must not be together. (2 marks) the novels must be together and the chemistry books must be together. (2 marks) the mathematics books must be together and the novels must not be together. (2 marks) (d) -END-​

Sagot :

MrRoyal

The number of ways of the books can be arranged are illustrations of permutations.

  • When the books are arranged in any order, the number of arrangements is 3628800
  • When the mathematics book must not be together, the number of arrangements is 2903040
  • When the novels must be together, and the chemistry books must be together, the number of arrangements is 17280
  • When the mathematics books must be together, and the novels must not be together, the number of arrangements is 302400

The given parameters are:

[tex]\mathbf{Novels = 3}[/tex]

[tex]\mathbf{Mathematics = 2}[/tex]

[tex]\mathbf{Chemistry = 5}[/tex]

(a) The books in any order

First, we calculate the total number of books

[tex]\mathbf{n = Novels + Mathematics + Chemistry}[/tex]

[tex]\mathbf{n = 3 + 2 + 5}[/tex]

[tex]\mathbf{n = 10}[/tex]

The number of arrangement is n!:

So, we have:

[tex]\mathbf{n! = 10!}[/tex]

[tex]\mathbf{n! = 3628800}[/tex]

(b) The mathematics book, not together

There are 2 mathematics books.

If the mathematics books, must be together

The number of arrangements is:

[tex]\mathbf{Maths\ together = 2 \times 9!}[/tex]

Using the complement rule, we have:

[tex]\mathbf{Maths\ not\ together = Total - Maths\ together}[/tex]

This gives

[tex]\mathbf{Maths\ not\ together = 3628800 - 2 \times 9!}[/tex]

[tex]\mathbf{Maths\ not\ together = 2903040}[/tex]

(c) The novels must be together and the chemistry books, together

We have:

[tex]\mathbf{Novels = 3}[/tex]

[tex]\mathbf{Chemistry = 5}[/tex]

First, arrange the novels in:

[tex]\mathbf{Novels = 3!\ ways}[/tex]

Next, arrange the chemistry books in:

[tex]\mathbf{Chemistry = 5!\ ways}[/tex]

Now, the 5 chemistry books will be taken as 1; the novels will also be taken as 1.

Literally, the number of books now is:

[tex]\mathbf{n =Mathematics + 1 + 1}[/tex]

[tex]\mathbf{n =2 + 1 + 1}[/tex]

[tex]\mathbf{n =4}[/tex]

So, the number of arrangements is:

[tex]\mathbf{Arrangements = n! \times 3! \times 5!}[/tex]

[tex]\mathbf{Arrangements = 4! \times 3! \times 5!}[/tex]

[tex]\mathbf{Arrangements = 17280}[/tex]

(d) The mathematics must be together and the chemistry books, not together

We have:

[tex]\mathbf{Mathematics = 2}[/tex]

[tex]\mathbf{Novels = 3}[/tex]

[tex]\mathbf{Chemistry = 5}[/tex]

First, arrange the mathematics in:

[tex]\mathbf{Mathematics = 2!}[/tex]

Literally, the number of chemistry and mathematics now is:

[tex]\mathbf{n =Chemistry + 1}[/tex]

[tex]\mathbf{n =5 + 1}[/tex]

[tex]\mathbf{n =6}[/tex]

So, the number of arrangements of these books is:

[tex]\mathbf{Arrangements = n! \times 2!}[/tex]

[tex]\mathbf{Arrangements = 6! \times 2!}[/tex]

Now, there are 7 spaces between the chemistry and mathematics books.

For the 3 novels not to be together, the number of arrangement is:

[tex]\mathbf{Arrangements = ^7P_3}[/tex]

So, the total arrangement is:

[tex]\mathbf{Total = 6! \times 2!\times ^7P_3}[/tex]

[tex]\mathbf{Total = 6! \times 2!\times 210}[/tex]

[tex]\mathbf{Total = 302400}[/tex]

Read more about permutations at:

https://brainly.com/question/1216161

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