If we put randomly 5 books of math, 6 of biology, 8 of history and 3 of literature. which the probability of the book of math are together?
My work:
Let $S:$"The set of solutions", then $|S|=22!$
Moreover,
Let $E:$"Book of math together" a event.
Let $M:$"The books of math" and we count that set as $|M|=1.$
Let $B:$"The other books" $|B|=17$
We need know of how many ways we have to order $M$ in the set of 18 books $(|B|+|M|)$
This is: $18$!.
Then we have $18!$ ways of order the books of math together.
In consequence, the probability of the book of math are together is: $\frac{18!}{22!}$
Is good the reasoning?
Best Answer
Assume that the books are individually distinguishable: i.e. that one math book (say) is different from another. That seems to be what you are assuming in your approach.
As you say, there are $22!$ ways in total.
There are $5!$ ways to arrange math books as a block. Once you have chosen one of those, there are $18!$ ways to arrange the $18$ objects consisting of the $17$ non-math books and the block of math books. Hence the answer is $$\boxed {\frac {5!\times 18!}{22!}}$$