Question

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

Answer 1

Step 1: Calculate the total number of permutations

The given word is “MISSISSIPPI”.

Here, the number of $\mathrm{M}$ is $1$, $\mathrm{I}$ is $4$, $\mathrm{S}$ is $4$ and $\mathrm{P}$ is $2$.

So, the total number of letters $=1+4+4+2$

$\Rightarrow$ the total number of letters $=11$

Thus, the total number of ways to arrange all the letters can be calculated as follows,

The total number of ways to arrange $11$ letters $=\frac{11!}{1!\times 4!\times 4!\times 2!}$

$\Rightarrow$ The total number of ways to arrange $11$ letters $=\frac{11\times 10\times 9\times 8\times 7\times 6\times 5\times 4!}{1\times 4!\times 4\times 3\times 2\times 1\times 2\times 1}$

$\Rightarrow$ The total number of ways to arrange $11$ letters $=34650$

Step 2: Calculate the number of permutations when I's comes together

Now, let us say that the $4\mathrm{I}\text{'}\mathrm{s}$ come together. And, then assume these $4\mathrm{I}\text{'}\mathrm{s}$ to be a single letter.

Then, the number of available letters $=11-4+1$

$\Rightarrow$ the number of available letters $=8$

Thus, the number of ways to arrange these $8$ letters can be calculated as follows,

The total number of ways to arrange $8$ letters $=\frac{8!}{1!\times 1!\times 4!\times 2!}$

$\Rightarrow$ The total number of ways to arrange $8$ letters $=\frac{8\times 7\times 6\times 5\times 4!}{1\times 1\times 4!\times 2\times 1}$

$\Rightarrow$ The total number of ways to arrange $8$ letters $=840$

Step 3: Calculate the required number of permutations

Now, the number of distinct permutations of the letters in “MISSISSIPPI” where the four I’s not come together can be calculated as,

The number of required permutations $=$ total number of ways to arrange $11$ letters $-$ total number of ways to arrange $8$ letters

$\Rightarrow$ The number of required permutations $=34650-840$

$\Rightarrow$ The number of required permutations $=33810$

Hence, in $33810$ distinct permutations of the letters in “MISSISSIPPI” the four I’s not come together.