Question

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

Answer 1

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
$=\frac{11!}{4!4!2!}$

$=\frac{11\times10\times9\times8\times7\times6\times5\times4!}{4!\times4\times3\times2\times1\times2\times1}$

$=\frac{11\times10\times9\times8\times7\times6\times5}{4\times3\times2\times1\times2\times1}$
$=34650$
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 $\frac{8!}{4!2!}$ 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 = $34650 - 840 = 33810$