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In how many of the distinct permutations of the letters in MISSISSIPPI do the four I’s not come together?

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In the given word MISSISSIPPI, I appears 4 times, S appears 4 times, P appears 2 times and M appears just once.
Therefore, number of distinct permutations of the letters in the given word 
= 11!/ 4!4!2!
= 11*10*9*8*7*6*5*4!/4!*4*3*2*1*2*1
= 34650
There are 4 Is in the given word. When they occur together, they are treated as a single object IIII for the time being. This single object together with the remaining 7 objects will account for 8 objects.
These 8 objects in which there are 4 Ss and 2 Ps can be arranged in 8!/4!2! ways i.e., 840 ways.
Number of arrangement where all Is occur together = 840
Thus, number of distinct permutations of the letters in MISSISSIPPI in which four Is do not come together 

= 34650-840 = 33810

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