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?

Rationale

There are 36 arrangements possible with the letters H, A, T, L, E, and W where the 1st, 2nd, and 4th letters must be L, E, and H in some order.

To find the total arrangements, we first determine the number of ways to arrange the letters L, E, and H in the specified positions. There are 3! (6) ways to arrange these three letters. The remaining letters A, T, and W can occupy the remaining positions, which can be arranged in 3! (6) ways as well. Therefore, the total arrangements are 6 (for L, E, H) multiplied by 6 (for A, T, W), giving us 36 arrangements.

A (32) YOUR ANSWER

This number does not accurately account for all possible arrangements of the letters. It fails to consider the full factorial combinations of the remaining letters after fixing L, E, and H in specific positions.

B (36) CORRECT

This is correct as it correctly calculates the arrangements of the letters with L, E, and H fixed in the 1st, 2nd, and 4th positions, allowing for the remaining letters A, T, and W to fill the other positions.

C (64) YOUR ANSWER

This choice overestimates the arrangements by incorrectly calculating the permutations available for the remaining letters. It may arise from mistakenly doubling the arrangements or misapplying the factorial principle.

D (120) YOUR ANSWER

This figure represents the total arrangements of all six letters without any restrictions. It does not apply to the specific requirement of fixing L, E, and H in designated positions.

E (720) YOUR ANSWER

This option also reflects the total arrangements of all six letters in a completely unrestricted form. Like option D, it does not consider the specific positional constraints for L, E, and H.

Conclusion

The correct calculation for the arrangements of the letters H, A, T, L, E, and W, given the constraints of having L, E, and H in the 1st, 2nd, and 4th positions, leads to 36 possible configurations. This involves fixing the positions of certain letters and permuting the others, demonstrating the application of combinatorial principles in a structured manner.

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?

There are 36 arrangements possible with the letters H, A, T, L, E, and W where the 1st, 2nd, and 4th letters must be L, E, and H in some order.

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