Question

Find the number of ways in which 5 persons A, B, C, D, E can be seated at a round table such that C and D must not sit together?
(a) 24
(b) 12
(c) 6
(d) 36

Answer 1

Hint: First of all, find the number of ways in which 5 persons will sit at a round table. We know that the ways in which n persons will sit at a round table is $\left( n-1 \right)!$. So, we are going to use this formula for finding the number of ways in which 5 persons will sit at a round table. Now, we will find the number of cases in which C and D will sit together by making C and D as a one person and then add this one person to the remaining three persons. So, we will find the number of ways in which 4 persons will sit at a round table then we will multiply the internal arrangement of two persons (C and D) which is 2!. For the final answer, we are going to subtract the number of ways when C and D are together from the ways when all the persons will sit at a round table.

Complete step by step solution:
We are going to find the number of ways in which 5 persons will sit at a round table by using the formula when n persons will sit at a round table which is equal to $\left( n-1 \right)!$.
Substituting the value of n as 5 in $\left( n-1 \right)!$ formula we get,
$\begin{align}
  & \Rightarrow \left( 5-1 \right)! \\
 & =4! \\
 & =4.3.2.1 \\
 & =24 \\
\end{align}$
Hence, we have found the total ways in which 5 persons can sit at a round table is 24.
Now, we are going to find the ways in which C and D are sitting together. For that, we are considering C and D as a one person and we are adding this one person with remaining 3 persons we get,
$\begin{align}
  & \Rightarrow 1+3 \\
 & =4 \\
\end{align}$
So, arranging 4 persons at a round table is equal to:
$\begin{align}
  & \left( 4-1 \right)! \\
 & =3! \\
 & =3.2.1 \\
 & =6 \\
\end{align}$
Now, we are going to multiply the internal arrangement of C and D. There are two possibilities in which C and D can be arranged which are:
CD, DC
So, multiplying 2 by 6 we get,
$\Rightarrow 12$
Hence, the number of ways in which 5 persons are arranged so that C and D are together is 12.
Subtracting the arrangement of 5 persons when C and D are together from the arrangement when in any way 5 persons can sit at a round table we get,
$\begin{align}
  & \Rightarrow 24-12 \\
 & =12 \\
\end{align}$
Hence, the number of ways of arranging 5 persons at a round table so that C and D won’t sit together is 12.
Hence, the correct option is (b).

Note: The mistake that could be possible in the above problem is that you might forget to multiply the internal arrangement of C and D while calculating the number of ways in which 5 persons will sit at a round table so make sure you won’t make this mistake.