15. In a word game, the 6 letters H, A, T, L, E, and W are to be arranged to form a word. The 1st, 2nd, and 4th letters must be L, E, and H in some order. How many arrangements, including those that form a word and those that do not, are possible?
Answer: B
There are 36 possible arrangements of the letters H, A, T, L, E, and W with the specified conditions.
To determine the total arrangements, we note that the 1st, 2nd, and 4th letters must be L, E, and H in some order, which can be arranged in 3! (6) ways. The remaining 3 letters (A, T, W) can occupy the 3rd, 5th, and 6th positions, which can be arranged in 3! (6) ways as well. Therefore, the total arrangements are 6 (for L, E, H) × 6 (for A, T, W) = 36.
A) 32
This option is incorrect because it underestimates the total number of arrangements. The arrangement calculation must account for the factorial permutations of both sets of letters, which amounts to 36, not 32.
B) 36
This is the correct option as it accurately reflects the total number of arrangements. The calculation considers the permutations of the letters constrained by the positions of L, E, and H, leading to the correct total of 36 arrangements.
C) 64
Selecting this option would imply that there are more arrangements than actually possible under the given constraints. The combinations do not allow for 64 distinct arrangements, as shown by the proper calculation of 36.
D) 120
This option overestimates the number of arrangements. While 120 may seem plausible if all letters were free to combine, the restrictions on L, E, and H greatly reduce the total combinations to 36.
E) 720
This option is incorrect as it assumes that all letters can be arranged without restriction. The specific conditions placed on the arrangement of L, E, and H significantly limit the total combinations, resulting in only 36 valid arrangements.
Conclusion
The correct answer is 36 arrangements, determined by the factorial permutations of the constrained and unconstrained letters. Other options fail because they miscalculate the number of valid configurations based on the established letter positions. Thus, option B stands as the definitive answer reflecting the constraints provided in the question.