$9$ different books are to be arranged on a book-shelf. $4$ of these books were written by Shakespeare, $2$ by Dickens and $3$ by Conrad. How many possible permutations are there if
(a) the books by Conrad are separated from each other?
(b) two books of Conrad are always together?
I want to know how can I solve part b. I have solved part 1 whose answer is
$7P3 \cdot 6!$
Note: I don't just need an answer. I need the idea.
Idea on b) (that's where you asked for):
You could find out how many arrangements exist in which books of Conrad are never together.
In all other arrangements two books of Conrad can be found that are together.
Start with finding out how many sums $a_1+a_2+a_3+a_4=6$ exist where $a_1,a_4$ (utmost left and utmost right non-Conrad books) are nonnegative integers and $a_2,a_2$ (non-Conrad books in between) are positive integers. You can use stars and bars for finding this. First step: find the number of sums $b_1+b_2+b_3+b_4=4$ where the $b_i$ are nonnegative integers.
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If it is requested that in the arrangement $2$ books of Conrad are consecutive and not $3$ books of Conrad are consecutive then we need a different approach.
Firstly there are $3$ ways to select two books. Secondly for the two selected ones there are $2$ orders. Thirdly the two books can be at the left of the single book or at the rigth, so a factor $2$ arises.
Now treat the two consecutive books of Conrad as one unit. To be found is the number of sums $c_1+c_2+c_3=6$ where $c_1,c_3$ are nonnegative integers (they represent the number on non-Conrad books at utmost left and utmost right) and $c_2$ (representing the non-Conrad books in between the Conrad units) is a positive integers. That comes to the same as finding the number of sums $d_1+d_2+d_3=5$ where the $d_i$ are nonnegative integers. With stars and bars we find $\binom72$ possibilities.
Then the non-Conrad books can be ordered on $6!$ ways.
Final result:$$3\times2\times2\times\binom72\times6!=181440 $$
If it concerns $2$ specific Conrad books (so no choice) then factor $3$ falls out and we arrive at $60480$.