Question

If $12$ persons are seated at a round table, what is the probability that two particular persons sit together?
A. $\dfrac{2}{{11}}$
B. $\dfrac{1}{{11}}$
C. $\dfrac{1}{{12}}$
D. $\dfrac{5}{{12}}$

Answer 1

Hint: First of all understand the question and use the formula for probability to get the required value. Also consider two particular persons as one as they always sit together on the round table. To get the number of ways a person can sit use the factorial of the number.

Complete step by step answer:
Here, we are given that $12$persons are seated on the round table. Therefore, the total number of ways in which these $12$ persons can sit along the round table can be given as $ = (12 - 1)! = 11!$ ways …. (A)
Now, we are given that two particular persons sit together so considering these two persons as one. There will be only $11$ persons and these $11$ can be seated along the round table by $(11 - 1)! = 10!$ ways …. (B)
These two particular persons can be arranged among themselves by $2!$ ways …. (C)
By using the values of the equation (B) and (C)
The total number of favorable outcomes can be given $ = 10! \times 2!$ ways … (D)
Now, the required probability can be given by taking the ratio of the favorable outcomes with the total number of possible outcomes.
By using the value of the equation (A) and (D)
Required probability $ = \dfrac{{10! \times 2!}}{{11!}}$
Expand the above expression –
Required Probability $ = \dfrac{{10! \times 2 \times 1}}{{11 \times 10!}}$
Common factors from the numerator and the denominator cancel each other.
Required Probability $ = \dfrac{2}{{11}}$

Hence, option A is the correct answer.

Note: Do not get confused between the terms factors and factorial and apply accordingly. Factorial is defined as the product of an integer along with the smaller integers below it till the number one. So, first consider all the possible ways of the arrangement and then place it in the formula.