Question

7 persons to be seated in a row. Probability that 2 particular persons to sit next to each other:
1. \[\dfrac{3}{7}\]
2. \[\dfrac{2}{7}\]
3. \[\dfrac{4}{7}\]
4. \[\dfrac{5}{7}\]

Answer 1

Hint: This question is a common case of permutations and combinations problems. Knowing the following formulas of probability and permutation and combinations would help you solve the question easily:
\[{ \Rightarrow ^n}{C_r} = \dfrac{{n!}}{{(n - r)!*r!}}\]
\[{ \Rightarrow ^n}{P_r} = \dfrac{{n!}}{{(n - r)!}}\]
\[ \Rightarrow P(E) = \dfrac{F}{T},\] where F are the Favourable cases and T are the total number of cases

Complete step-by-step answer:
So, Let’s start the question by calculating the total number of cases which is the arrangement of 7 people in a row,
\[ \Rightarrow T{ = ^7}{P_7}\]
Using the formula of permutations, we get,
\[ \Rightarrow T = \dfrac{{7!}}{{(7 - 7)!}}\]
Simplifying the expression,
\[ \Rightarrow T = 7! - - - - (i)\]
Now, Let’s calculate the favourable cases where two people always sit together,
Now, to approach this part, let us tie those people together, therefore now we have to arrange only six people with the condition that those two people can also arrange among themselves,
This can be achieved by:
\[ \Rightarrow F{ = ^6}{P_6}{ \times ^2}{P_2}\]
Doing calculations,
\[ \Rightarrow F = \dfrac{{6!}}{{(6 - 6)!}} \times \dfrac{{2!}}{{(2 - 2)!}}\]
\[ \Rightarrow F = 6! \times 2! - - - - (ii)\]
The probability of the event is given by
\[ \Rightarrow P = \dfrac{F}{T}\]
Now, using \[(i)\]and \[(ii)\]we get,
\[ \Rightarrow P = \dfrac{{6! \times 2!}}{{7!}}\]
Cancelling the common factors in numerator and denominator and doing the calculations, we get,
\[ \Rightarrow P = \dfrac{2}{7}\]
Thus, option(2) is the correct answer.
So, the correct answer is “Option 2”.

Note: This question can be tedious if you are not well versed with topics like permutations, combinations, and probability. Be sure to apply the correct formula, keeping in mind the calculations. We must know how to calculate the factorials. Care should be taken while handling the calculative steps so as to get to the correct answer.