Help on clarification regarding permutation

[vedd21]

So my syllabus has started permutation and combination recently and I seem to understand its basic functions and when to use it until I come across a particular question which goes like this...

Here are the answers

So I got my answer for (a) through the product of 7! x 4!.
But question (b) made me doubt my understanding towards permutation,
So step I, I assume the total arrangement for the other books are 6! since the total of Biology and Chemistry books are 6.
Second step, since the spaces in between the 6 books are 7 in total(including the spaces beside the first and last book) the 4 Mathematics book can be placed in between any of the 7 spaces.
So I came up with, 6! x 7 x 6 x 5 x 4
-6! which is from the other 6 books
-7 x 6 x 5 x 4 which is the space remaining to be filled by the other Mathematics book after the 1st book is placed until the 4th.
The answer I got is 604800.(through calculator)
But its far from the original answer.
I discussed with my teacher and she told me that I should find the total arrangement of all 10 books which is 10! and substract it with the answer I obtained at question (a) which is the product of 7! x 4! (120960).
Now I used this method for other questions and they worked fine for example,

I used the same method and the one my teacher suggested and they both have the same answer.
5! x 6 x 5 (my method)
-5! being the total letters without vowel
-Start with 6 because there are 6 spaces in between 5 consonants(including spaces beside the first consonant and the last)
-6 x 5 being the space remaining after the 1st vowel is filled in between consonants and the 2nd afterwards
7! - 6! x 2!(my teacher's method)
-7! being the total arrangement of all the letters
-6! x 2! being the arrangement if the vowels are together
-6! since 2 vowels are considered one. Thus, instead of 7 its 6.
-2! being the arrangement of the vowels together
Both equations gave the exact same answer which is 3600.

I know I am missing something but I just couldn't figure out what is it and my teacher told me I should just stick to using her method but I wanna know what do you guys think?
Thank you for your time.

 

[Dr.Peterson]

I think this is a matter of interpretation of the English, and I think you are right. Was this written by a native English speaker?

To say "all the Mathematics books are separated" clearly means to me what it does to you: no two Math books are together.

What your teacher is calculating is the number of arrangements in which not all of the Math books are together, so that at least one is separate from the others. That is entirely different from what was written. Equating the two is either bad English or bad logic.

Never let a teacher tell you to "just stick to using her method". You need to understand.

 

[Dr.Peterson]

I used the same method and the one my teacher suggested and they both have the same answer.
5! x 6 x 5 (my method)
-5! being the total letters without vowel
-Start with 6 because there are 6 spaces in between 5 consonants(including spaces beside the first consonant and the last)
-6 x 5 being the space remaining after the 1st vowel is filled in between consonants and the 2nd afterwards
7! - 6! x 2!(my teacher's method)
-7! being the total arrangement of all the letters
-6! x 2! being the arrangement if the vowels are together
-6! since 2 vowels are considered one. Thus, instead of 7 its 6.
-2! being the arrangement of the vowels together
Both equations gave the exact same answer which is 3600.

I just looked at the second example for the first time, and realized why in that case both answers are the same:

When there are only two vowels, "no two vowels together" is equivalent to "not all vowels together", because "all" and "two" are the same.

In the first problem, there are four math books, and "no two math books together" is not equivalent to "not all math books together", because in the latter case there can be 3 math books together, but not in the former case.

 

[vedd21]

I think this is a matter of interpretation of the English, and I think you are right. Was this written by a native English speaker?

To say "all the Mathematics books are separated" clearly means to me what it does to you: no two Math books are together.

What your teacher is calculating is the number of arrangements in which not all of the Math books are together, so that at least one is separate from the others. That is entirely different from what was written. Equating the two is either bad English or bad logic.

Never let a teacher tell you to "just stick to using her method". You need to understand.

I see. No these questions were not written by native English speaker.

However if that is the case, if I were to encounter questions like this again, which method would be appropriate to use? I want to know if my method can still be used in questions like "total arrangement such that not all objects are together/at least one is separated from the others".

 

[Dr.Peterson]

I see. No these questions were not written by native English speaker.

However if that is the case, if I were to encounter questions like this again, which method would be appropriate to use? I want to know if my method can still be used in questions like "total arrangement such that not all objects are together/at least one is separated from the others".

What I recommend when there are issues of interpretation (as opposed to just different methods for solving the same problem, which is not what is happening here) is to state your interpretation of the problem, paraphrasing it in much the same way I did, before showing the work. This way it is clear that your work is correct for the problem as you interpreted it.

That is not always possible, and the person grading may disagree with your anyway; in those cases, you just need to be able to defend your answer.

 

[vedd21]

What I recommend when there are issues of interpretation (as opposed to just different methods for solving the same problem, which is not what is happening here) is to state your interpretation of the problem, paraphrasing it in much the same way I did, before showing the work. This way it is clear that your work is correct for the problem as you interpreted it.

That is not always possible, and the person grading may disagree with your anyway; in those cases, you just need to be able to defend your answer.

I see.Thank you for your suggestion.