Question

In how many ways can the letters of the word $"\mathrm{PERMUTATIONS}"$ be arranged so that there are always $4$ letters between $\text{'}\mathrm{P}\text{'}$ and $\text{'}\mathrm{S}\text{'}$?

Answer 1

Step 1: Form the expression representing required number of ways

The given word is $"\mathrm{PERMUTATIONS}"$.

Also given that there should always be $4$ letters between $\text{'}\mathrm{P}\text{'}$ and $\text{'}\mathrm{S}\text{'}$.

Since there are a total of $12$ letters in the word $"\mathrm{PERMUTATIONS}"$.

So, the number of places for each letter are $1,2,3,4,5,6,7,8,9,10,11,12$.

According to the question, the possible places of $\text{'}\mathrm{P}\text{'}$ and $\text{'}\mathrm{S}\text{'}$ are $\left(1,6\right),\left(2,7\right),\left(3,8\right),\left(4,9\right),\left(5,10\right),\left(6,11\right)$ and $\left(7,12\right)$.

So, the number of possible ways $=7$,

Also, the places of $\text{'}\mathrm{P}\text{'}$ and $\text{'}\mathrm{S}\text{'}$ can be interchanged,

So, the number of total possible ways $=2\times 7=14$

After fixing the places of $\text{'}\mathrm{P}\text{'}$ and $\text{'}\mathrm{S}\text{'}$, the remaining $10$ places can be filled with the remaining $10$ letters. So,

The required number of ways $=\frac{{}^{10}P_{10}}{2!}$

Step 2: Simplify the above expression

Using the rules of permutation, the above expression can be simplified as,

$\Rightarrow$ The required number of ways $={}^{10}P_{10}\times \frac{1}{2!}$

$\Rightarrow$ The required number of ways $=\frac{10!}{\left(10-10\right)!}\times \frac{1}{2!}$ $\mathrm{[}\mathrm{\because }\mathrm{}{}^{\mathrm{n}}\mathrm{P}_{\mathrm{r}}\mathrm{}\mathrm{=}\mathrm{}\frac{\mathrm{n}\mathrm{!}}{\left(n-r\right)\mathrm{!}}\mathrm{]}$

$\Rightarrow$ The required number of ways $=\frac{10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{1}\times \frac{1}{2\times 1}$

$\Rightarrow$ The required number of ways $=1814400$

Hence, the letters of the word $"\mathrm{PERMUTATIONS}"$ can be arranged in $1814400$ ways so that there are always $4$ letters between $\text{'}\mathrm{P}\text{'}$ and $\text{'}\mathrm{S}\text{'}$.