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?

Filo · Accepted Answer

In the given word MISSISSIPPI, there are total

11

letters.

I

appears

4

times,

S

appears

4

times,

P

appears

2

times.

Therefore, number of distinct permutations of the letters in the given word

= \frac{11 !}{4 ! \times 4 ! \times 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

I^{'} s

in the given word.

When they occur together, they are considered as a single object . This single object (letter) together with the remaining

7

objects(letters) will give

8

objects.

These objects in which there are

4 S^{'} s

and

2 P^{'} s

can be arranged in

\frac{8 !}{4 ! 2 !}

ways i.e.,

840

ways.

Number of arrangements where all

I^{'} s

occur together

= 840

Thus, number of distinct permutations of the letters in

M I SS I SS I PP I

in which four

I^{'} s

do not come together

= 34650 - 840 = 33810

Question Text	In how many of the distinct permutations of the letters in MISSISSIPPI do the four I's not come together?
Answer Type	Text solution:1
Upvotes	150

Text solutionVerified