Question

Four men and three women are standing in a line for railway tickets. The probability of standing them in an alternate manner is
(A) $\dfrac{1}{84}$
(B) $\dfrac{1}{7}$
(C) $\dfrac{1}{35}$
(D) $\dfrac{1}{33}$

Answer 1

Hint: For answering this question we will use the given information in the question and extract the total possible number of ways in which the members can be arranged in a line and the possible number of ways in which they can be arranged in an alternative manner and then obtain the probability of standing them in an alternative manner using $\dfrac{\text{ number of possible ways of them standing in an alternative manner}}{\text{Total number of ways possible}}$.

Complete step-by-step solution
Now considering the question we have four men and three women standing in a line for a railway ticket.
There are 7 members in total.
The total possible number of ways in which the 7 members can be arranged in a line is $7!$.
The possible number of ways in which they can be arranged in an alternative manner is $3! \times 4!$. Hence the probability of standing them in an alternative manner is $\dfrac{\text{ number of possible ways of them standing in an alternative manner}}{\text{Total number of ways possible}}$.
By substituting these in the place we will have $\dfrac{3!\times 4!}{7!}$.
After expanding we will have $\dfrac{\left( 3\times 2 \right)\times \left( 4\times 3\times 2 \right)}{7\times 6\times 5\times 4\times 3\times 2}$.
By simplifying this we will have $\dfrac{1}{7\times 5}$.
Hence we will have the final answer as $\dfrac{1}{35}$.
Hence we can conclude that the probability of standing four men and three women in an alternate manner in a straight line for a railway ticket is $\dfrac{1}{35}$.
Hence we will mark the option C as correct.

Note: While answering questions of this type we should take care of the calculations. And we should make a note that the possible number of ways is different from the probability. If we had forgotten to extract probability from the number of ways and tried to answer we would be in a mess. And if we had made a calculation mistake and taken $\dfrac{3!\times 4!}{7!}$ as $\dfrac{1}{84}$ we will mark option A as correct which is wrong.