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  • A shelf contains 6 different books, of which 3 are novels, 2 are poetry collections, and 1 is a biography. In how many ways can the books be arranged on the shelf if book

A shelf contains 6 different books, of which 3 are novels, 2 are poetry collections, and 1 is a biography. In how many ways can the books be arranged on the shelf if books of the same genre must be placed together?

Option: 1

52


Option: 2

96


Option: 3

72


Option: 4

86

 


Answers (1)

best_answer

To calculate the number of ways the books can be arranged on the shelf while keeping books of the same genre together, we can treat each genre (novels, poetry collections, and biography) as a single entity and arrange them first. Then, we can arrange the three groups of genres on the shelf.

Let's break down the calculation step by step:

Calculate the number of ways to arrange the novels as a group:

Since there are 3 novels, we can arrange them among themselves in 3! (3 factorial) ways.

Calculate the number of ways to arrange the poetry collections as a group:

Similarly, since there are 2 poetry collections, we can arrange them among themselves in 2! (2 factorial) ways.

Calculate the number of ways to arrange the biography:

As there is only 1 biography, it can be arranged in 1 way.

Calculate the number of ways to arrange the three groups of genres on the shelf:

Now, we have three groups (novels, poetry collections, and biography) that need to be arranged on the shelf. Since these groups are distinct from each other, we can arrange them in 3! ways.

Multiply the results from steps 1, 2, 3, and 4 to get the total number of arrangements:

Multiply the number of arrangements for each step:

\mathrm{3 ! \times 2 ! \times 1 \times 3 !=6 \times 2 \times 1 \times 6=72 .}

Therefore, there are 72 ways to arrange the books on the shelf if books of the same genre must be placed together.

Hence option 3 is correct.

 

Posted by

Pankaj

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