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

Question

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

BrentGMATPrepNow · Accepted Answer

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

Answer:  E

RELATED VIDEO
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Skywalker18 · Answer

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
Answer E

rohit8865 · Answer

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)

Ans E

ccooley · Answer

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: M S  S  S  S PP
a 'different' arrangment: M S  S  S  S PP

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.

Zoser · Answer

What about this approach?

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?

GmatIvyPrep · Answer

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!}

Pritishd · Answer

Hi  GmatIvyPrep

This does not seem to be the correct answer as you may have overlooked the constraint of keeping all vowels (letter I in this case) together.
 \frac{11!}{4!4!2!}  gives us the total ways of arranging the 11 letters of the word MISSISSIPPI.

Warm Regards,
Pritish

Kinshook · Answer

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

1. M-1
2. I - 4
3. S - 4
4. P - 2

Vowels = I 
If 4 I s are to be together then IIII string should be considered a single letter

1. M-1
2. IIII - 1
3. S - 4
4. P - 2

No of arrangements = 8! / (4! * 2!) = 840 ways

IMO E

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

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

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