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? 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?
 A: 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 $$\boxed {\frac {5!\times 18!}{22!}}$$
A: https://faculty.math.illinois.edu/~ekirr/page/teaching/math461/quiz1solution.pdf
hehe... I found the first question beneficial ..see if it helps :)
A: Total possibilities is, as you said, $22!$.
if we number the places of the books from $1$ to $22$, then there will be $
18$ possibilities for the math books to be together without counting the permutations.
from $ N^° 1$ to $N^° 5$
from $2$ to $6$
.....
and from $18$ to $22$.
The answer to your question is
$$\frac {18\times 5!}{22!} .$$
A: Is this the original wording? The grammar is a bit off. I assume the question is what the probability of all the math books being together is. 
Your answer is almost right. You're on the right track of thinking of the math books as one block that can be moved together, but you didn't notice that there are 5! ways of creating this block. So it should be 5!18!/22!
Notice that this is equal to 18/(22 choose 5); another way of looking at it is that there are 18 different choices of where to put the first math book. Once you choose a place for the first math book, there are 22 choose 5 ways of splitting the 22 locations into "math book locations" and "not math book locations", and only one of those corresponds to five math books together starting at the location you chose. So the probability is 1/(22 choose 5) for each location for the first math book, and a total of 18/(22 choose 5).
