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# In how many different ways can the letters of the word MISSISSIPPI be

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Math Expert
Joined: 02 Sep 2009
Posts: 47977
In how many different ways can the letters of the word MISSISSIPPI be  [#permalink]

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08 Apr 2016, 02:47
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45% (medium)

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70% (01:32) correct 30% (02:42) wrong based on 149 sessions

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In how many different ways can the letters of the word MISSISSIPPI be arranged if the vowels must always be together?

A. 48
B. 144
C. 210
D. 420
E. 840

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Re: In how many different ways can the letters of the word MISSISSIPPI be  [#permalink]

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08 Apr 2016, 21:21
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The word MISSISSIPPI has 4 I , 4S , 2P and M .
The only vowel present in the word is I and we have to consider 4 I's as a single group

Number of different ways = 8!/(4!*2!)
=(8*7*6*5)/2
=840
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Re: In how many different ways can the letters of the word MISSISSIPPI be  [#permalink]

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26 Jun 2017, 12:18
1
Bunuel wrote:
In how many different ways can the letters of the word MISSISSIPPI be arranged if the vowels must always be together?

A. 48
B. 144
C. 210
D. 420
E. 840

vowel (I) together can be arranged with other 7 letters in 8! ways
4 no. vowels themselves can be arranged in 4! ways

different ways = 8!*4!/( 4!*4!*2!) = 840---(4I's , 4 S's , 2 P's)

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Re: In how many different ways can the letters of the word MISSISSIPPI be  [#permalink]

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26 Jun 2017, 13:18
2
2
Bunuel wrote:
In how many different ways can the letters of the word MISSISSIPPI be arranged if the vowels must always be together?

A. 48
B. 144
C. 210
D. 420
E. 840

Ooh, a combinatorics problem with some fairly large numbers. The best technique for these is to mentally break them down into simpler problems. Otherwise, the solutions tend to sort of look like magic tricks - sure, you can use the formula, but how are you supposed to know to use that formula, and not one of the many similar-looking ones?

There are four vowels, and they're all the same. Let's set those vowels aside: (IIII)

Now we have the letters MSSSSPP. How many ways can just those letters be arranged? Well, if they were all different, we could arrange them in 7x6x5x4x3x2x1 = 7! ways. However, they aren't all different. For instance, four of the letters are (S). I'm going to color them to demonstrate why that matters:

one arrangement: MSSSSPP
a 'different' arrangment: MSSSSPP

We counted both of those arrangements, but we actually don't want to. All of the Ss look the same - they aren't actually different colors - so we want to make those arrangements the same. Because there are 4x3x2x1 = 24 ways to order the different Ss, we want every set of 24 arrangements where the Ss are in the same place, to just count as 1 arrangement, instead. So, we can divide out the extra possibilities by dividing our total by 24 (or 4!):

7!/4!

Do the same thing with the two Ps. We still have twice as many arrangements as we need, since we've counted as if the two Ps were different, but they're actually the same. So, divide the total by 2:

7!/(4! x 2)

Now we have to put the (IIII) letters back in. They all have to go together. Start by looking at one of the arrangements of the other letters: PMSSPSS. Where can the four Is go? There are 8 places where we can put them:

IIIIPMSSPSS
PIIIIMSSPSS
PMIIIISSPSS
PMSIIIISPSS

etc.

So, for each arrangement, we have to multiply by 8, to account for the eight possible ways to put the vowels back in. Here's the final answer:

(8 x 7!) / (4! x 2) = (8 x 7 x 6 x 5 x 4 x 3 x 2) / (4 x 3 x 2 x 2) = 4 x 7 x 6 x 5 = 4 x 210 = 840.
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Re: In how many different ways can the letters of the word MISSISSIPPI be  [#permalink]

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26 Jun 2017, 13:40
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Top Contributor
2
Bunuel wrote:
In how many different ways can the letters of the word MISSISSIPPI be arranged if the vowels must always be together?

A. 48
B. 144
C. 210
D. 420
E. 840

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!)]

-------NOW ONTO THE QUESTION!!!-----------------

First "glue" the 4 I's together to create ONE character: IIII (this ensures that they stay together)
So, basically, we must determine the number of arrangements of M, S, S, S, S, P, P and IIII
There are 8 characters in total
There are 4 identical S's
There are 2 identical P's
So, the total number of possible arrangements = 8!/[(4!)(2!)]
= 840

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Re: In how many different ways can the letters of the word MISSISSIPPI be  [#permalink]

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04 Dec 2017, 09:31

Stage 1: Lets place the letters without the 4I's:
MSSSSPP
7!/(4!2!)= 105 ways

Stage 2: Now lets place the 4I's
-m-s-s-s-s-p-p-
We can place th4 4I's in 8 ways

Total ways= 105*8= 840 ways.

Is this a correct approach?
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Joined: 25 May 2017
Posts: 10
GMAT 1: 750 Q49 V44
Re: In how many different ways can the letters of the word MISSISSIPPI be  [#permalink]

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04 Dec 2017, 10:08
Rearranging letters has a simple formula:

Total number of letters
*DIVIDED BY*
(2)! for every letter repeated twice / (3)! for every letter repeated thrice and so on...

Mississippi has:

4 s's
4 i's
2 p's

Total number of letters: 11!

So the answer is $$\frac{11!}{4!4!2!}$$
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Re: In how many different ways can the letters of the word MISSISSIPPI be &nbs [#permalink] 04 Dec 2017, 10:08
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