Question

{"given":"8 chairs numbered 1–8. 2 women choose from chairs 1–4; 3 men choose from remaining chairs.","key_observation":"Since each person occupies a distinct (labelled) chair, order matters and permutations must be used. After 2 women pick from chairs 1–4, there are $4 - 2 + 4 = 6$ chairs left for the men.","option_analysis":[{"label":"(A)","text":"$^6C_3 \\times {^4C_2}$","verdict":"incorrect","explanation":"$^4C_2 \\times {^6C_3} = 6 \\times 20 = 120$, but this uses combinations (ignoring order), whereas chairs are labelled so arrangements must be counted using permutations, giving $^4P_2 \\times {^6P_3} = 1440$."},{"label":"(B)","text":"$^4P_2 \\times {^4P_3}$","verdict":"incorrect","explanation":"Women correctly choose from 4 chairs in $^4P_2$ ways, but men must choose from the 6 remaining chairs (not just 4), so the second factor should be $^6P_3$, not $^4P_3$."},{"label":"(C)","text":"$^4C_2 \\times {^4C_3}$","verdict":"incorrect","explanation":"Both factors use combinations (which ignore seating order) and the second factor incorrectly uses only 4 chairs instead of the 6 remaining; the correct count is $^4P_2 \\times {^6P_3} = 1440$."},{"label":"(D)","text":"None of these","verdict":"correct","explanation":"The correct count is $^4P_2 \\times {^6P_3} = 12 \\times 120 = 1440$, which matches none of options (A), (B), or (C)."}],"answer":"(D)","formula_steps":[]}