How many ways can French and Spanish books be arranged on a shelf?

[mtingt]

Homework Statement

there are 7 different french books and 7 different Spanish books, how many ways are there to arrange them on a shelf
a. books of the same language must be group together, French on left and Spanish on Right?
b. French and Spanish books must alternate in the grouping, beginning with a French book?

I tried doing 7!x7! for both of them but i don't think i am right?

I have no idea how to approach this

 

[haruspex]

I tried doing 7!x7! for both of them but i don't think i am right?

Sounds right to me.

 

[MadAtom]

for the first one 7!x7! seems right, but for the 2nd one I think (not sure...!) it's :

7C1X7C1 X 6C1X6C1 X 5C1X5C1 X 4C1X4C1 X 3C1X3C1 X 2C1X2C1 X 1C1X1C1

 

[CAF123]

for the first one 7!x7! seems right, but for the 2nd one I think (not sure...!) it's :

7C1X7C1 X 6C1X6C1 X 5C1X5C1 X 4C1X4C1 X 3C1X3C1 X 2C1X2C1 X 1C1X1C1

I think an argument could go: there are 14 choices for the first book, (French or Spanish). There are then 7 choices for the next book (If first was French, this one must be Spanish), then 6 choices for next, (has to be French), then 6 choices for next (has to be Spanish)...and so on. In this Q, the French book is first so what you wrote is correct.

 

[Ray Vickson]

I think an argument could go: there are 14 choices for the first book, (French or Spanish). There are then 7 choices for the next book (If first was French, this one must be Spanish), then 6 choices for next, (has to be French), then 6 choices for next (has to be Spanish)...and so on. In this Q, the French book is first so what you wrote is correct.

Another argument: for each arrangement of the French books, leave a space between successive books and fill those spaces with the Spanish books, one book per space.

RGV

 

[haruspex]

for the 2nd one I think it's :
7C1X7C1 X 6C1X6C1 X 5C1X5C1 X 4C1X4C1 X 3C1X3C1 X 2C1X2C1 X 1C1X1C1

How is that different from 7!x7!?
The two obviously have the same answer. Either way, there is a fixed set of 7 positions that can be taken by the French books, and another fixed set of 7 that can be taken by the Spanish, independently.

 

[Ray Vickson]

How is that different from 7!x7!?
The two obviously have the same answer. Either way, there is a fixed set of 7 positions that can be taken by the French books, and another fixed set of 7 that can be taken by the Spanish, independently.

It's not different; it's just another argument that the OP may, or may not, prefer.

RGV

 

[haruspex]

It's not different; it's just another argument that the OP may, or may not, prefer.
RGV

I was replying to MadAtom, who wrote:

the first one 7!x7! seems right, but for the 2nd one I think (not sure...!) it's :​

Seems to me MadAtom implied 7!x7! was wrong for the second question.

 

[MadAtom]

Seems to me MadAtom implied 7!x7! was wrong for the second question.

I thought so, but the result is the same... sorry.