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

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Homework Help Overview

The problem involves arranging 7 different French books and 7 different Spanish books on a shelf, with specific conditions for the arrangement. The first condition requires all French books to be grouped on the left and Spanish books on the right. The second condition requires the books to alternate, starting with a French book.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the factorial approach (7!x7!) for the first condition and question its applicability to the second condition. Some suggest combinatorial reasoning involving choices for each position in the alternating arrangement.

Discussion Status

There is an ongoing exploration of different arguments and interpretations regarding the arrangement methods. Some participants express uncertainty about the correctness of their approaches, while others provide alternative reasoning that aligns with the factorial method.

Contextual Notes

Participants are navigating potential misunderstandings about the equivalence of different approaches to the same problem. The discussion reflects a mix of confidence and uncertainty regarding the methods proposed.

mtingt
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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
 
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mtingt said:
I tried doing 7!x7! for both of them but i don't think i am right?
Sounds right to me.
 
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
 
MadAtom said:
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.
 
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CAF123 said:
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
 
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MadAtom said:
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.
 
haruspex said:
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
 
Ray Vickson said:
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.
 
haruspex said:
Seems to me MadAtom implied 7!x7! was wrong for the second question.

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

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