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  • A set of 8 books includes 3 novels, 2 biographies, and 3 self-help books. In how many different ways can the books be arranged on a shelf if the books of the same genre m

A set of 8 books includes 3 novels, 2 biographies, and 3 self-help books. In how many different ways can the books be arranged on a shelf if the books of the same genre must be kept together?

 

Option: 1

2,903,040 

 


Option: 2

4,752,856

 


Option: 3

6,789,452

 


Option: 4

1,268,953


Answers (1)

best_answer

To determine the number of different ways the books can be arranged on the shelf, we'll treat the books of the same genre as a single entity. We'll consider the groups of novels, biographies, and self-help books as distinct items.

Number of distinct items = Number of distinct groups + Number of distinct books within the groups

\mathrm{ \begin{aligned} & =3+2+3 \\ & =8 \end{aligned} }

Now, we can arrange these 8 distinct items on the shelf. The total number of different arrangements can be calculated as 8 factorial (8!).

Total number of arrangements =8 !

However, within each group of books (novels, biographies, self-help books), the books can be arranged among themselves.

So, we need to consider the permutations within each group as well.

Number of arrangements within each group:

Novels: 3 factorial (3!) since there are 3 novels.

Biographies: 2 factorial (2!) since there are 2 biographies.

Self-help books: 3 factorial (3!) since there are 3 self-help books.

Therefore, the total number of different arrangements is:

\mathrm{ \begin{aligned} & \text { Total number of arrangements }=8 ! \times(3 ! \times 2 ! \times 3 !) \\ & 8 !=8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1=40,320 \\ & 3 !=3 \times 2 \times 1=6 \\ & 2 !=2 \times 1=2 \end{aligned} }

Substituting these values back into the expression:

40,320 \times(6 \times 2 \times 6)=40,320 \times 72=2,903,040

Therefore, the number of different ways the books can be arranged on the shelf is 2,903,040.

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shivangi.shekhar

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