Question

There are 5 different French books and 5 different Spanish books. How many ways are there to arrange them on a shelf if

a) Books of the same language must be grouped together, French on the left, Spanish on the right?
b) French and Spanish books must alternate in the grouping, beginning with a French book?

Answer 1

a) There are 14,400 ways to arrange the books if books of the same language must be grouped together. b) There are 5,760 ways to arrange the books if French and Spanish books must alternate in the grouping.

Step-by-step explanation:

a) To determine the number of ways to arrange the books of the same language grouped together, you need to calculate the permutations of the French books and the permutations of the Spanish books separately, and then multiply the results. There are 5 French books, so there are 5! = 120 ways to arrange them. Similarly, there are 5 Spanish books, so there are 5! = 120 ways to arrange them. The total number of ways to arrange the books of the same language grouped together is 120 x 120 = 14,400.

b) To determine the number of ways to arrange the French and Spanish books alternatingly, you need to consider the order of the books. You will start with a French book, so there are 5 choices for the first book. After that, you will alternate between French and Spanish, so there are 4 choices for the second book, and then 4 choices for the third book, and so on. The total number of ways to arrange the books alternatingly is 5 x 4 x 4 x 3 x 3 x 2 x 2 x 1 x 1 = 5760.

Answer 2

b because french and Spanish are alternate soo they grouped and spanish goes second and begin with french book