# In how many ways can the letters of the word PERMUTATIONS be arranged if the There are always $4$ letters between $P$ and $S$?

$\begin{array}{1 1}(A)\;2550100\\(B)\;2540100\\(C)\;2560111\\(D)\;2560000\end{array}$

There are 12 letters to be arranged in 12 place.
There 12 letters are to filled in 12 places shown
P may be filled up at place No.1,2,3,4,5,6,7 and consequently S may be filled up at place No.6,7,8,9,10,11,12 having four places in between.Now P and S may be filled up in 7 ways
Similarly,S and P may be filled of in 7 ways
Thus P and S or S and P can be filled up in 7+7=14 ways
Now remaining 10 places can be filled by E,R,M,U,T,A,T,I,O,N in $\large\frac{10!}{2!}$ ways
$\therefore$ No of ways in which 4 letters occur between P and S
$\Rightarrow \large\frac{10!}{2!}$$\times 14 \Rightarrow \large\frac{10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{2\times 1}$$\times 14$
$\Rightarrow 2540100$
Hence (B) is the correct answer.