Question

There are 4 different French books and 4 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

Good question. Lets look at part a first.

So we want to know how many different ways we can pick the order for the 4 french books, then how many different ways we can pick the order of the 4 spanish books. Are you familiar with permutations and the formula for it?

Answer 2

how would i set letter b up?

Answer 3

The same way you set up the first one. If you MUST alternate french and spanish books, then it's essentially the same problem, it's just that rather than do all the french ones, then all the spanish ones you interleave them. But you still have the same number of ways you can change their orders.

Answer 4

sorry that was to the other response! but its deleted now. yes im familiar with permutations but i don't know how to set a and b up

Answer 5

Ok, well we have 2 tasks to do, to complete this job. We must first arrange all the french books. How many are there? And how many will we put on the shelf?

Answer 6

(this is for part a btw)

Answer 7

there are 4 french books and we will put all four on the shelf

Answer 8

Correct!

So how many different ways can you arrange 4 french books on a shelf if you use all 4 of them?

Answer 9

16

Answer 10

\[4P4 = 4! = 4*3*2*1 = 4*6 = 24\]

Answer 11

Ok, so we've done the first part of the job. The second part is to arrange the spanish books. How many are there? And how many different ways can we arrange those?

Answer 12

24

Answer 13

Right. So now we have 2 different tasks to do to complete this job (part a) We have 24 different ways to do the first task, and 24 different ways to do the second task.

By the product rule then, we have 24*24 different ways we can do the whole job.

Answer 14

So how many different ways can we arrange the shelf doing the french books first, and the spanish books second?

Answer 15

576

Answer 16

Correct.

Answer 17

Ok, now there's a couple of ways we can do part b.

Answer 18

But most of them are tedious. ;p

Answer 19

is it just the same answer as before?

Answer 20

It is actually, because once you choose the order of the french books and the order of the spanish books, you just have to alternate them instead of keeping them together. The number of different ways you order them doesn't change.

Answer 21

perfect!!!!