$3$ math books, $5$ English books, $4$ science books and a dictionary are to be placed on a student's shelf so that the books of each subject remain together.
In how many different ways can the books be arranged?
$$3!\space5!\space4!\space4!$$
$3!$ ways to arrange the math books, $5$ for English, $4$ for science, and $4!$ ways to arrange all these different subject books around.
In how many of these will the dictionary be next to the maths books?
I'm not quite sure how to do the second question. Can someone please illustrate how I could find out the number of arrangements in this case.. A photo that illustrate this would be nice.. I can't quite visualize it too well.
Best Answer
For the placement of subjects, there are $4$ slots that each subject can be placed in. With the added constraint that the dictionary and math books next to each other, the math books and dictionary will be considered together as a single "unit". In other words, there will be a math/dictionary slot, which along with science and English makes for a total of $3$ slots rather than $4$.
In addition, there are also 2 possible permutations among the math books and the dictionary, so the total number is:
$$3!\space5!\space4!\space3!\space2!$$