Question

Thirteen persons take their places at a round table$,$ show that it is five to one against two particular persons sitting together.

Answer 1

$⟹$ To prove $=P\left(\overline{A}\right)/P\left(A\right)=5:1$
For Round Table arrangement
If there are $n$ persons,
Possible arrangement $⟹(n-1)!$
For thirteen persons $⟶(13-1)!=12!$
For particular two person consider them one.
Now, for twelve persons $⟶(12-1)!=11!$
Expected $⟹11!2!$ ($2!⟶$ Arranging them together )
Total Outcomes $=12!$
Probability person sit together $=\frac{11!2!}{12!}=\frac{1}{6}$
$P\left(\overline{A}\right)=\frac{5}{6}$
$\therefore P\left(\overline{A}\right)/P\left(A\right)=5:1$