Deepak Scored 45->99%ile with Bounce Back Crack Course. You can do it too!

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

Question:

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

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

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

$=\frac{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 !}{4 ! \times 4 \times 3 \times 2 \times 1 \times 2 \times 1}$

$=\frac{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1 \times 2 \times 1}$

$=34650$

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 $4 \mathrm{Ss}$ 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