Question

If all the letters of the word "SUCCESS" are written down at random in a row , then the probability that no two C's and no two S's occur together is

• A. $\frac{2}{35}$
• B. $\frac{8}{35}$
• C. $\frac{2}{7}$
• D. None of these

Answer 1

The correct option is B $\frac{8}{35}$

The number of ways to write the letters of the word "SUCCESS" is $\frac{7!}{3!\times 2!}=420$.

Now we are to find the ways of writing the letters of that word such that no two C's and no two S's occur together.

To do this first we are to count then ways to form the word in this format "_U_C_C_E_ " such that no two S occur together.

To do this we have ${}^5{C}_3\times \frac{4!}{2!}=120$ ways where no two S occur together but there may the case that two C occurs together.

Two avoid this we are to avoid the words like _U_CC_E_, this types of words are in ${}^4{C}_3\times 3!=24$ numbers.

So the number of ways to write the given word where no two C's and no two S's occur together $=120-24=96$ and the required probability $=\frac{96}{420}=\frac{8}{35}$.