Question

In how many ways may 3 books be placed next to each other on a shelf?

Short answer

There are \(6\) different ways to arrange the 3 books on the shelf.

Step-by-step solution

Identify the permutation formula

To find the number of ways to arrange given objects, we use the permutation formula: nPr = n! / (n-r)!, where n is the total number of objects, r is the number of objects being arranged, and nPr is the number of permutations.

Apply the permutation formula to the given problem

In our problem, we have 3 books to be arranged in a row (n=3), and we need to arrange all of them (r=3). Thus, the formula now becomes: 3P3 = 3! / (3-3)!

Calculate the factorials

Now, we need to calculate the factorials: 3! = 3 × 2 × 1 = 6 (3-3)! = 0! = 1 (by definition, 0! = 1)

Substitute the factorials back into the formula

Now, we can substitute the calculated factorials back into our formula: 3P3 = 6 / 1

Calculate the number of permutations

Finally, we can complete the calculation and find the number of permutations: 3P3 = 6 So, there are 6 different ways to arrange the 3 books on the shelf.