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 marked 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 must know what the difference between ${}^m{P_n}{\text{ and }}{}^m{C_n}$ is. Here ${}^m{P_n}{\text{ and }}{}^m{C_n}$ have the difference that ${}^m{C_n}$ represents the number of ways by which we can select $n$ out of $m$ things while ${}^m{P_n}{\text{ }}$ is the representation of selecting as well as arranging $n$ out of $m$ things. So here we can proceed by finding arrangements of men and women separately and then multiply them.

Complete step by step solution:
Here we are given that eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. So we have a total of eight chairs that are to be occupied. As it is told that firstly, the women choose the chairs from amongst the chairs marked 1 to 4. So we can say that we need to arrange two women on the four chairs. So as we have to find the total number of arrangements and not only the selections, so we will use the permutation instead of combination.
So total chairs to occupy women are $4$
Number of women $= 2$
So total arrangements are ${}^4{P_2}$
Now we have two chairs filled with women. So we have left number of chairs as $8 - 2 = 6$
So on these chairs we need to arrange men who are $3$ in number.
So total ways of arranging $3$ men on $6$ chairs is ${}^6{P_3}$
As both these events are happening simultaneously so we need to multiply both the cases for getting the required result.
So total arrangements that can be made are ${}^4{P_2} \times {}^6{P_3}$

Hence D is the correct option.

Note:
Here the student can make the mistake at a point where he needs to subtract two chairs for the remaining chairs where men are to be arranged. So there will be $6{\text{ not 4}}$ chairs where men need to be seated. So one must be careful while reading the statement carefully.