Question

How many ways can 72 different books be arranged in a rack if two particular books cannot remain together?

a) (72!)
b) (71!)
c) (72! - 2!)
d) (72! - 2 times 70!)

Answer 1

Calculate the total number of books arrangements (72!) and subtract from it the scenarios where two particular books are together (2 times 70!), to find the ways they can be arranged without being together.

Step-by-step explanation:

To determine the number of ways 72 different books can be arranged on a rack where two specific books are not together, consider two scenarios:

1. Total arrangements without any restriction, which is the factorial of 72 (72!).
2. Arrangements where the two particular books are together. Treat these two books as one unit, giving us 71 units to arrange (71!), and then multiply by 2! because the two books can be arranged in 2 ways within this unit (2 × 71!).

The number of arrangements where the two specific books are not together is the total arrangements minus the arrangements where they are together, which is 72! minus 2 × 71!. Therefore, the correct option is d) (72! - 2 times 70!), as 2 × 71! simplifies to 2 × 70! because 71! = 71 × 70!.