Question

Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First, the women choose the chairs from amongst the chairs numbered 1 to 4, and then, the men select the chairs from amongst the remaining. The number of possible arrangements is:
(a) ${}^6{C_3} \times {}^4{C_2}$
(b) ${}^4{P_2} \times {}^4{P_3}$
(c) ${}^4{C_2} + {}^4{P_3}$
(d) None of these

Answer 1

Here, we have to find the number of possible arrangements. First, we will find the number of arrangements in which the two women can take 2 chairs from the chairs 1 to 4. Then, we will find the number of arrangements in which the three men can take 3 chairs from the remaining chairs. Finally, we will multiply the number of arrangements in which the men can take the 3 chairs, and the number of arrangements in which the women can take the 2 chairs from chairs numbered 1 to 4, to get the required number of possible arrangements.

Formula used:
The number of permutations of selecting $r$ objects from a set of $n$ objects (where order is important) is given by ${}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$.

Complete step by step solution:
We can observe that the two women choose one chair each from the four chairs numbered 1 to 4.
Since the women are distinguishable (different), we will use permutations instead of combinations to consider the order as well.
The number of arrangements in which the 2 women can take 2 chairs out of the 4 chairs is given by ${}^4{P_2}$.
Now, the remaining number of chairs is 6 chairs, since the women only occupied two chairs from the first 4 chairs.
Thus, the 3 men can choose to occupy one chair each from the remaining 6 chairs.
Since the men are distinguishable (different), we will use permutations instead of combinations to consider the order as well.
The number of arrangements in which the 3 men can take 3 chairs out of the 6 chairs is given by ${}^6{P_3}$.
Now, we will multiply the number of arrangements in which the men can take the 3 chairs, and the number of arrangements in which the women can take the 2 chairs from chairs numbered 1 to 4.
Therefore, we get
Number of possible arrangements $= {}^4{P_2} \times {}^6{P_3}$
Therefore, we get the total number of possible arrangements as ${}^4{P_2} \times {}^6{P_3}$.
Thus, the correct option is option (d).

Note: We should understand why we used permutations and not combinations. We know that each man and woman are different. Suppose that the two women are A and B. Then, if A takes chair 1, and B takes chair 2, we get one arrangement. If B takes chair 1, and A takes chair 2, we get another arrangement. If we used combinations, it would include the above two situations as the same arrangement, since it does not consider the order. Therefore, we used permutations.