Question

{"given":"The word GARDEN contains 6 letters: G, A, R, D, E, N. Among these, there are 2 vowels (A and E) and 4 consonants (G, R, D, N). We need to find arrangements where vowels appear in alphabetical order. ","key_observation":"When arranging letters with a constraint on the order of specific letters, we use the principle that exactly half of all arrangements will have those letters in the desired order. Since vowels A and E can be arranged in 2! = 2 ways, and we want only the alphabetical order (A before E), we divide the total arrangements by 2.","option_analysis":[{"label":"(A)","text":"120","verdict":"incorrect","explanation":"This would be the result if we were arranging only 5 letters (5! = 120), but GARDEN has 6 letters, so this is too small."},{"label":"(B)","text":"480","verdict":"incorrect","explanation":"This is not related to any standard permutation calculation for this problem. It's neither the total arrangements nor the correct constrained arrangements."},{"label":"(C)","text":"360","verdict":"correct","explanation":"Total arrangements of GARDEN = 6! = 720. Since vowels A and E can be in any relative order, exactly half will have A before E (alphabetical order). Therefore, arrangements with vowels in alphabetical order = 720/2 = 360."},{"label":"(D)","text":"240","verdict":"incorrect","explanation":"This is one-third of the total arrangements (720/3 = 240), which doesn't correspond to any relevant constraint in this problem."}],"answer":"(C)","formula_steps":[]}