SOLUTION: In how many ways we can arrange the letters in the word 'MISSISIPI' so that all S's come together and 2 I's come together . the spelling of the word has been given like missisipi

Question 894461: In how many ways we can arrange the letters in the word 'MISSISIPI' so that all S's come together and 2 I's come together . the spelling of the word has been given like missisipi
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
your original string of letters is missisipi.
rearrange the terms so all the occurrences of s are together and all the occurrences of i are together.
you will get miiiisssp.
replace every 3 occcurrences of s with s and replace every 2 occurrences of i with i.
you will get miisp
the number of possible arrangements of miisp will be equal to 5! / 2! = (5*4*3*2*1) / (2*1) which becomes 5*4*3 which becomes 60.
the number of possible arrangements of miisp will be equal to 60.
that should beyour answer.
there are 60 possible arrangements.

there are too many to go through to confirm, but we can use a simpler example to see if the formula is correct.
start with missisipi
let's drop an s and drop a p.
we are left with missiii
we want to group all occurrences of s into one s and every pair of occurrences of i into one i.
after we do that, we get:
miis
this is 4 characters and the number of possible arrangements will be 4! / 2! which is equal to 4*3 = 12
this is a little bit more manageable so lets see what we get.

the 12 possible arrangements are:

iims
iism
imis
imsi
isim
ismi
miis
misi
msii
siim
simi
smii

now we need to expand to see what we get:

we will replace every occurrence of s with ss and every occurrence of i with ii

we get:

iiiimss
iiiissm
iimiiss
iimssii
iissiim
iissmii
miiiiss
miissii
mssiiii
ssiiiim
ssiimii
ssmiiii

every pair of occurrences of i are together and all occurrences of s are together so we have achieved what we want to achieve.

this confirms the formula is good and we can be reasonably confident it is correct when we apply it to the larger problem as we did on top.

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