Question

In how many ways $5$ persons can sit at a round table, if two of the person do not sit together?

Answer 1

Suppose The $5$ persons are $A,B,C,D,E$

Total number of ways that $5$ persons can sit at a round table $\left(m\right)=\frac{{}^5{p}_5}{5}=24.$

There’s $1$ way to sit $A$ because all seats look alike at an empty table. Then, there are $2$ ways to seat $B$, $3$ ways to seat $C$, $2$ ways to seat $D$, and one empty seat for $E$,

Say total ways so that two of the person do not sit together $\left(n\right)=1\times 2\times 3\times 2\times 1=12$

Hence, the resultant $=m-n=24-12=12.$

In $12$ ways, $5$ persons can sit at a round table, if two of the person do not sit together.