Given ten letters : K,K,K,S,S,S,S,S,S,S. Find number of ways to arrange given ten letters such that no K should be there between two S?

Question

Given ten letters : K,K,K,S,S,S,S,S,S,S.

How many ways to arrange given ten letters such that no letter "K" between two letters "S", example : "KKSSSSSSSK", "KKKSSSSSSS", "SKKSSSSSSK", etc.

I am confused.

I have calculate ways to arrange 10 given letters, $$\dfrac{10!}{3!\cdot 7!}=120 \text{ ways.}$$ Now I want to calculate the complement of "to arrange given ten letters such that no "K" between two "S" ", that is "there is letter "K" between two letters "S"". If I calculate how many ways : $$8\cdot \dfrac{7!}{2!\cdot 5!}=168\text{ ways.}$$

That is impossible, negative number. $$120-168=-48\text{ ways.}$$

How to solve that combinatorics problem? Please help me :(

Answer 1

You don't require the rule for complements to do this. Just count the ways to arrange

• arrange KKK,S,S,S,S,S and placing remaining S one each at the ends.
• arrange KK,S,S,S,S,S,S,S and placing the remaining K at either end.