How many ways can it be arranged on a shelf?

Let A, B, C, D, and E represent the 5 different books
So, we want to arrange the following 20 letters: AAAABBBBCCCCDDDDEEEE

-------ASIDE---------------------------------------------
When we want to arrange a group of items in which some of the items are identical, we can use something called the MISSISSIPPI rule.
It goes like this:

If there are n objects where A of them are alike, another B of them are alike, another C of them are alike, and so on, then the total number of possible arrangements = n!/[(A!)(B!)(C!)....]

So, for example, we can calculate the number of arrangements of the letters in MISSISSIPPI as follows:
There are 11 letters in total
There are 4 identical I's
There are 4 identical S's
There are 2 identical P's
So, the total number of possible arrangements = 11!/[(4!)(4!)(2!)]

----------ONTO THE QUESTION--------------------------

GIVEN: AAAABBBBCCCCDDDDEEEE
There are 20 letters in total
There are 4 identical A's
There are 4 identical B's
There are 4 identical C's
There are 4 identical D's
There are 4 identical E's
So, the total number of possible arrangements = 20!/[(4!)(4!)(4!)(4!)(4!)]
= 20!/[(4!)^5]

Answer: C

Cheers,
Brent

formula : The number of ways in which MN different items can be divided equally into M groups, each containing N objects and the order of the groups is important is = (mn)!/(n!)^m

20!/(4!)^5= C

We have 5 books A,B,C,D,E and 4 copies of each. Therefore we have A1,A2,A3,A4,B1,B2,B3,B4,C1,C2,C3,C4,D1,D2,D3,D4,E1,E2,E3,E4 = 20 BOOKS

The way to rearrange 20 items is by 20x19x18x17x16....=20!

Lets keep in mind that we rearrange our 4 copies of each book by 4x3x2x1=4!

Therefore we have 5 items repeated 4 times and we need to account for the copies resulting in 20! divided by 4!x4!x4!x4!x4!

Result is option C = 20!/(4!)^5

20!/((4!)^5)

Answer - C
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Could someone explain me the rationale behind this formula ?

Posted from GMAT ToolKit

Look it as number of ways of arranging AAAA BBBB CCCC DDDD EEEE books where A,B,C,D,E repeat 4 times.
hence 20!/ 4!^5

20! / ((4!)^5)

this division is done to avoid repetitions.

Lets say we have to figure out number of arrangements for A,B1,B2. (where B1=B2) . total arrangements for 3 letters is 3!.

A-B1-B2
A-B2-B1 - duplicate as B1=B2
B1-A-B2
B2-A-B1 - duplicate as B1=B2
B1-B2-A
B2-B1-A - duplicate as B1=B2

so to avoid duplicates we need to divide the total arrangements/ (number of similar items)! = 3!/2!

thnx for the formula. please tell me what is the formula if order is not important
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C

It's a basic formula

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