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.
$$^4{P_2} \times {\,^4}{C_3}$$
C.
$$^4{C_2} + {\,^4}{P_3}$$
D.
none of these
Answer :
none of these
Solution :
$$\overline 1 \,\,\overline 2 \,\,\overline 3 \,\,\overline 4 \,\,\overline 5 \,\,\overline 6 \,\,\overline 7 \,\,\overline 8 $$
Two women can choose two chairs out of 1, 2, 3, 4, in $$^4{C_2}$$ ways and can arrange themselves in 2! ways. Three men can choose 3 chairs out of 6 remaining chairs in $$^6{C_3}$$ ways and can arrange themselves in 3! ways
∴ Total number of possible arrangements are
$$^4{C_2} \times 2!\,\, \times {\,^6}{C_3} \times \,3! = {\,^4}{P_2} \times {\,^6}{P_3}$$