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Counting question

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Amrabdelnaby Master | Next Rank: 500 Posts Default Avatar
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Counting question

Post Fri Nov 13, 2015 11:04 am
Could you please explain the below question? Smile

How many different four letter words can be formed (the words need not be meaningful) using the letters of the word "MEDITERRANEAN" such that the first letter is E and the last letter is R?

A. 59

B. 11! / (2!*2!*2!)

C. 56

D. 23

E. 11! / (3!*2!*2!*2!)

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joseph2017 Newbie | Next Rank: 10 Posts
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Post Sun Jul 03, 2016 6:05 am
Hi Brent,

The question clearly says ' How many different 4 letter words can be formed', so shouldn't the arrangement be E-(11*10)-R .

Joseph.

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Post Fri Nov 13, 2015 1:58 pm
Amrabdelnaby wrote:
Could you please explain the below question? Smile

How many different four letter words can be formed (the words need not be meaningful) using the letters of the word "MEDITERRANEAN" such that the first letter is E and the last letter is R?

A. 59
B. 11! / (2!*2!*2!)
C. 56
D. 23
E. 11! / (3!*2!*2!*2!)
Since the first and last letters are FIXED, we have E _ _ _ _ _ _ _ _ _ _ _ R
So, we need to determine the number of ways to arrange the remaining 11 letters (M,E,D,I,T,R,A,N,E,A,N) in the 11 spaces.

----------------------------------------------------------------------------------
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!)]
----------------------------------------------------------------------------------

In this question, we need to arrange 11 letters: M,E,D,I,T,R,A,N,E,A,N
There are 11 letters in total
There are 2 identical E's
There are 2 identical A's
There are 2 identical N's
So, the total number of possible arrangements = 11!/[(2!)(2!)(2!)]

Answer: B

Here are some related questions:
- http://www.beatthegmat.com/p-c-t274535.html
- http://www.beatthegmat.com/permutation-with-the-word-tennessee-t281906.html
- http://www.beatthegmat.com/strings-t284175.html
- http://www.beatthegmat.com/nemesis-t270113.html

Cheers,
Brent

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