Question

Eight chairs are numbered from $1$ to $8.$ Two women and three men wish to occupy one chair each. First, the women choose the chairs among the chairs marked from $1$ to $4,$ then the men select the chairs amongst the remaining. The number of possible arrangments is

• A. ${}^4{P}_2\cdot {}^6{P}_3$
• B. ${}^6{C}_3\cdot {}^4{C}_2$
• C. ${}^4{C}_2\cdot {}^4{P}_3$
• D. ${}^4{P}_2\cdot {}^4{P}_3$

Answer 1

The correct option is A ${}^4{P}_2\cdot {}^6{P}_3$

Two women can choose two chairs out of four in ${}^4{C}_2$ ways. The two women can also arrange themselves in $2!$ ways.

Similarly, the three men can choose the chairs in ${}^6{C}_3$ ways and can arrange themselves in $3!$ ways.

Hence, the required number of arrangements
$=({}^4{C}_2\times 2!)\times ({}^6{C}_3\times 3!)={}^4{P}_2\cdot {}^6{P}_3$