Question

If a party of $\text{n}$ persons sit at a round table, then the odds against two specified individuals sitting next to each other are

• A. $\text{2:(n-3)}$
• B. $\text{(n-3):2}$
• C. $\text{(n-2):2}$
• D. $\text{2:(n-2)}$

Answer 1

The correct option is B

$\text{(n-3):2}$

Finding the probability if the odds against two specified individuals sitting next to each other are:

The number of ways, in which $\text{“n”}$ number of persons sit in a round table$\text{= (n-1)!}$

Favorable cases $\text{= 2! (n-2)!}$

Hence, the required probability $\text{=}\frac{\text{2! (n-2)!}}{(n-1)!}$

$\text{=}\frac{\text{2}}{(n-1)}$

We know that, The odds against $\text{= (1-p : p)}$

$\text{=}\left(1-\left(\frac{2}{(\text{n}-1)}\right)\right)˸\frac{2}{(\text{n}-1)}$

$\text{= (n-3):2}$

Therefore, the odd against are $\text{(n-3):2}$

Hence, option $\left(2\right)$ is correct.