SOLUTION: There are 7 different large books, 5 equal medium ones and 3 equal small ones. In different ways can they be lined up on a shelf if the same sizes are to be placed together?

Question 1191664: There are 7 different large books, 5 equal medium ones and 3 equal small ones. In different ways can they be lined up on a shelf if the same sizes are to be placed together?
Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

Let's consider a related problem: we have 3 books labeled S, M and L (for small medium and large) and we want to arrange them.

There are 3! = 3*2*1 = 6 ways to do this
If small goes first
SML
SLM
Or if medium goes first
MSL
MLS
Or if large goes first
LSM
LMS

Once we have a particular permutation of L,M,S set up, we'll then consider the sub-permutations for each block.
The L block consists of 7 different large books we'll arrange together. There are 7! = 7*6*5*4*3*2*1 = 5040 ways to arrange just the large books.

For the M block, we have 5! = 5*4*3*2*1 = 120 ways to arrange those medium sized books.

The S block has 3! = 3*2*1 = 6 ways to arrange those books.

To summarize:

There are 6 ways to arrange S, M, and L
There are 5040 ways to arrange the large books
There are 120 ways to arrange the medium sized books
There are 6 ways to arrange the small books

We'll multiply these items out. Why multiplication? Well consider forming a table with 6 rows and 5040 columns (of course this is just a thought exercise and not something you actually have to draw out). There would be 6*5040 different inner cells to represent the number of permutations when considering just the first two items of the bullet list above. This is then extended to account for all the items above.

So we have 6*5040*120*6 = 21,772,800 different ways to arrange the books such that the sizes stick together, and that order matters.

Answer: 21,772,800
This is a little under 21.8 million

RELATED QUESTIONS
a student has 2 small books, five medium sized books, and three large ones. If all the... (answered by checkley77)

Find the number of ways 5 large books, 4 medium-size books, and 3 small books can be... (answered by ikleyn)

Find the number of ways in which 5 large books, 4 medium-size books and 3 small books can (answered by fractalier)

16 books are to be lined up on the shelf. If 6 of the books are identical math books and... (answered by math_tutor2020)

find the number of ways in which 5 large books, 4 medium size books and 3 small books can (answered by filboi)

In how many ways can 4 large books, 3 medium- size books, and 2 small books, be placed on (answered by srinivas.g)

A group of 8 teenagers are lined up to buy tickets to a rock concert. How many possible... (answered by richwmiller)

There are 4 different mathematics books and 5 different science books in how many ways... (answered by ikleyn)

Find the number of ways 5 large books and 3 small books can be placed on a shelf where... (answered by checkley79)