The probability of all the same type of books not being together on a shelf

Masaru

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The questions are as follows:

On a shelf there are 4 different mathematics books and 8 different English books.

i) The books are to be arranged so that the mathematics books are together. In how many different ways can this be done?


Answer: 4!*9! = 8 709 120

I did not have any problem with the question and answer above, but the problem is the next one:

ii) What is the probability that all the mathematics books are NOT together.

I would interpret it as the complement of "all the mathematics books are together", so my working would be, using the result of i) above,

1 -
8 709 120/12! = 54/55

However, the textbook says that the correct answer is 9/30.

Please tell me what's wrong with my working and why the answer is 9/30.

Also, even if we interpret the question as "none of the mathematics books are next to each other", a correct working would be:
(9P4)*8!/12! = 14/55, which is still different to the answer provided in the textbook.

I would much appereciate it if someone can help me to see what's wrong with my working.

Thank you.
 
The questions are as follows:

On a shelf there are 4 different mathematics books and 8 different English books.

i) The books are to be arranged so that the mathematics books are together. In how many different ways can this be done?


Answer: 4!*9! = 8 709 120

I did not have any problem with the question and answer above, but the problem is the next one:

ii) What is the probability that all the mathematics books are NOT together.

I would interpret it as the complement of "all the mathematics books are together", so my working would be, using the result of i) above,

1 -
8 709 120/12! = 54/55

However, the textbook says that the correct answer is 9/30.

Please tell me what's wrong with my working and why the answer is 9/30.

Also, even if we interpret the question as "none of the mathematics books are next to each other", a correct working would be:
(9P4)*8!/12! = 14/55, which is still different to the answer provided in the textbook.

I agree with your answers. Is it possible that you looked at the wrong answer?

I am assuming that the problem was originally in English and you copied the entire problem, exactly as given. As you note, the wording is a little ambiguous, so a difference in wording could make it erroneous.
 
Last edited:
Thank you very much for your confirmation that this textbook is wrong!


I agree with your answers. Is it possible that you looked at the wrong answer?

I am assuming that the problem was originally in English and you copied the entire problem, exactly as given. As you note, the wording is a little ambiguous, so a difference in wording could make it erroneous.

No, I did not look at the wrong answer.
I double and triple checked it and the text says clearly that the answer is 9/30.

It is written in Enlgish and I copied the entire problem as it is, so there is no room for me to make any mistake.

I often come across answers that I cannot make sense of in this textbook and actually they are wrong, so I think that this is simply another error - as is usual with this "badly proofread" textbook!

Thank you very much for your confirmation that this answer is wrong.
 
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