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

In the given word MISSISSIPPI:

• I appears 4 times
• S appears 4 times
• P appears 2 times
• M appears once

Therefore, number of distinct permutations of the letters in the given word = (11!)/(4! × 4! × 2! × 1!) = 34650

Now, when the 4 I’s occur together, they are treated as a single object (IIII). This single object together with the remaining 7 objects will account for 8 objects.

These 8 objects in which there are 4 S’s and 2 P’s can be arranged in (8!)/(4! × 2! × 1! × 1!) = 840 ways.

Number of arrangements where all I’s occur together = 840

Thus, number of distinct permutations of the letters in MISSISSIPPI in which four I’s do not come together = 34650 − 840 = 33810