Question

Out of 10 person sitting at a round table, two persons are selected at random then the probability that they are not adjacent to each other is
A. $\dfrac{5}{{12}}$
B. $\dfrac{7}{{12}}$
C. $\dfrac{5}{7}$
D. $\dfrac{7}{9}$

Answer 1

Hint: First, we learn the meaning of the terms permutation and combination which are important topics in probability.
In Probability, the permutation is the process of arranging the outcomes in order. Here, the order must be followed to arrange the items.
In Probability, the term combination refers to the process of selecting the outcomes in which the order does not matter. Here, the order to arrange the items is not followed.
Formula used:
\[\text{Probability of an event} = \dfrac{\text{Number of outcomes}}{\text{Total number of outcomes}}\]

Complete step-by-step solution:
It is given that ten persons are sitting at the round table.

Here, the first person has the choice of choosing all seats, and the second person can choose the remaining nine seats. Similarly, the third can choose the remaining eight seats and it goes on. The last one has to choose the remaining last one seat.
That is the total number of ways in which $10$ persons can sit at a round table $ = \left( {10 - 1} \right)!$
$ = 9!$ways
First, we shall calculate the probability that two persons are adjacent to each other.
Let the two persons be A and B.
Then eight people are there.
The number of ways in which eight persons can sit together is $ = \left( {9 - 1} \right)!$
$ = 8!$ ways
Here A and B can interchange their seats in $2!$ ways.
The required number of ways will be $2! \times 8!$
The probability that two persons are adjacent to each other$ = \dfrac{\text{Number of outcomes}}{\text{Total number of outcomes}}$
$ = \dfrac{{2! \times 8!}}{{9!}}$
$
= \dfrac{{2! \times 8!}}{{9 \times 8!}} \\
= \dfrac{2}{9} $
Hence, the probability that two persons are not adjacent to each other $ = 1 - \dfrac{2}{9}$
$ = \dfrac{{9 - 2}}{9} \\
   = \dfrac{7}{9} $
Therefore, the probability that two persons are not adjacent to each other is $\dfrac{7}{9}$. So, option (D) is correct.

Note: The formula to find the permutation is as follows.
\[{}_n{P_r} = n(n - 1)(n - 2).......(n - r + 1)\]
$ = \dfrac{{n!}}{{(n - r)!}}$ (! Is a mathematical symbol called the factorial)
Where $n$ denotes the number of objects from which the permutation is formed and $r$ denotes the number of objects used to form the permutation.
Now, the formula to calculate the combination is as follows.
${}_n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
Where $n$ denotes the number of objects from which the combination is formed and $r$ denotes the number of objects used to form the combination.