SOLUTION: in how many ways we can arrange the letters in the word 'MISSISIPI' so that all S's come together and no I's come together

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Question 891657: in how many ways we can arrange the letters in the word 'MISSISIPI' so that all S's come together and no I's come together
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
I'll assume that MISSISSIPPI was mispelled. It has 4 S's and 2 P's.
If you meant "MISSISIPI", there are so few you can just list them all
easier than calculating them, for it's just the 6 permutations of
M,P,(SSS) with 4 I's separating them:


IMIPISSSI
IMISSSIPI
IPIMISSSI     
IPISSSIMI     
ISSSIMIPI
ISSSIPIMI

Assuming you meant the correct spelling of the state:

%22%22-%22%22

The number of ways the four S's can come together

The letters of MISSISSIPPI arranged in alphabetical order is

I,I,I,I,M,P,P,S,S,S,S

The number of ways the four S's can come together is the number of
distinguishable permutations of these 8 things, where the (SSSS) is
considered as a single "thing".

I,I,I,I,M,P,P,(SSSS)

Since there are 4 indistinguishable I's and 2 indistinguishable P's,
 the number is:

8%21%2F%284%212%21%29=840

From this 840 we must subtract the number of ways 2 or more I's 
can come together.

First we will calculate the number of distinguishable arrangements of

M,P,P,(SSSS) with no I's, and then we'll calculate how many ways we 
can insert the 4 I's with 2 or more together. Then we'll multiply
those two numbers together.  

Again in the distinguishable arrangements of M,P,P,(SSSS), the 
(SSSS) is considered as just one thing.

Since there are 2 indistinguishable P's, the number is:

4%21%2F%282%21%29 = 12

An example would be PMP(SSSS). Let's put a space before and after 
each letter or "thing" to indicate feasible places to insert the I's.

  _P_M_P_(SSSS)_
            
So there are 5 places to insert the four I's.

Case 1.  All four I's come together (IIII). 
         That's 5 ways.

Case 2.  The I's are split 3 and 1, (III) and (I).
         Place the (III) any of 5 ways
         Place the (I) in any of the 4 remaining places
         That's 5*4 or 20 ways.
Case 3.  The I's are split 2 and 2. (II) and (II)
         Choose the 2 places for them to go in C(5,2) or 10 ways
         That's 10 ways. 
Case 4.  The I's are split 2,1 and 1.  (II),(I), and (I)
         Choose the places for the two single (I)'s C(5,2) = 10 ways
         Choose the place for the (II) as any of the remaining 3 ways
         That's 10*3 or 30 ways.

For those four cases that's 5+20+10+30 = 65 ways.

We multiply this 65 by the 12 distinguishable arrangements of M,P,P,(SSSS),  

That's 65*12 or 780 ways.

That's the number which we must subtract from the 840.

Final answer = 840 - 780 = 60 distinguishable permutations.

Here they all are, computer generated, 10 rows of 6 each:

IMIPIPISSSS    IMIPIPSSSSI   IMIPISSSSIP   IMIPISSSSPI   IMIPPISSSSI   IMIPSSSSIPI
IMISSSSIPIP    IMISSSSIPPI   IMISSSSPIPI   IMPIPISSSSI   IMPISSSSIPI   IMSSSSIPIPI
IPIMIPISSSS    IPIMIPSSSSI   IPIMISSSSIP   IPIMISSSSPI   IPIMPISSSSI   IPIMSSSSIPI
IPIPIMISSSS    IPIPIMSSSSI   IPIPISSSSIM   IPIPISSSSMI   IPIPMISSSSI   IPIPSSSSIMI
IPISSSSIMIP    IPISSSSIMPI   IPISSSSIPIM   IPISSSSIPMI   IPISSSSMIPI   IPISSSSPIMI
IPMIPISSSSI    IPMISSSSIPI   IPPIMISSSSI   IPPISSSSIMI   IPSSSSIMIPI   IPSSSSIPIMI
ISSSSIMIPIP    ISSSSIMIPPI   ISSSSIMPIPI   ISSSSIPIMIP   ISSSSIPIMPI   ISSSSIPIPIM
ISSSSIPIPMI    ISSSSIPMIPI   ISSSSIPPIMI   ISSSSMIPIPI   ISSSSPIMIPI   ISSSSPIPIMI
MIPIPISSSSI    MIPISSSSIPI   MISSSSIPIPI   PIMIPISSSSI   PIMISSSSIPI   PIPIMISSSSI
PIPISSSSIMI    PISSSSIMIPI   PISSSSIPIMI   SSSSIMIPIPI   SSSSIPIMIPI   SSSSIPIPIMI

Edwin