Carcass wrote:
There are 4 copies of 5 different books. In how many ways can they be arranged on a shelf?
A) \(\frac{20!}{4!}\)
B) \(\frac{20!}{5(4!)}\)
C) \(\frac{20!}{(4!)^5}\)
D) \(20!\)
E) \(5!\)
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