How many ways can it be arranged on a shelf?

Question

There are 4 copies of 5 different books. In how many ways can they be arranged on a shelf?

A) 20!/4!
B) 20!/5(4!)
C) 20!/(4!)^5
D) 20!
E) 5!

BrentGMATPrepNow · Accepted Answer

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

subhashghosh · Answer

20!/((4!)^5)

Answer - C

sudhir18n · Answer

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

Loki2612 · Answer

Could someone explain me the rationale behind this formula ?

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arzad · Answer

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

shashankp27 · Answer

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

Spidy001 · Answer

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!

LalaB · Answer

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

blink005 · Answer

C

It's a basic formula

bumpbot · Answer

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How many ways can it be arranged on a shelf?

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How many ways can it be arranged on a shelf?

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