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Question
In how many of the distinct permutations of the letters in MISSISSIPPI do the four I’s not come together?
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Solution
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 xx 10 xx 9 xx 8 xx 7 xx 6 xx 5 xx 4!)/(4! xx 4 xx 3 xx 2 xx 1 xx 2 xx 1)`
= `(11 xx 10 xx 9 xx 8 xx 7 xx 6 xx 5)/(4 xx 3 xx 2 xx 1xx 2 xx 1)`
= 34650
There are 4 Is in the given word. When they occur together, they are treated as a single object 
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 arrangements 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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