Probability of books not next to each other in a shelf

[ineedhelp33]

Question:

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 none of the mathematics books are next to each other?

The working given for this question is ((8!)*(9P4))/(12!), but could anyone explain what is the logic behind this working?

 

[Deleted member 4993]

Question:

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 none of the mathematics books are next to each other?

The working given for this question is ((8!)*(9P4))/(12!), but could anyone explain what is the logic behind this working?

How many ways can you stack 12 books , WITHOUT any restrictions?

 

[ineedhelp33]

How many ways can you stack 12 books , WITHOUT any restrictions?

12!, but what about the ((8!)*(9P4)) part?

 

[Dr.Peterson]

ii) What is the probability that none of the mathematics books are next to each other?

The working given for this question is ((8!)*(9P4))/(12!), but could anyone explain what is the logic behind this working?

You have to count the number of ways to arrange the books so that the math books are separated by others.

One way to make such an arrangement is to first arrange the 8 other books (in how many ways?), and then pick 4 places to put the math books. How many potential sites are there to put a math book, after arranging the others?