To find the number of arrangements of the letters in the word "honest" with specified conditions, we need to follow a systematic approach.
Understanding the Problem
- The word "honest" consists of 6 distinct letters: H, O, N, E, S, T.
- The conditions require that H always precedes E, and E always precedes T.
Calculating Total Arrangements Without Conditions
- The total arrangements of the 6 letters can be calculated using the factorial of the number of letters:
- Total arrangements = 6! = 720.
Applying the Conditions
- We must consider the constraints that H must come before E, and E must come before T.
- We can treat the trio (H, E, T) as a block that maintains the order H < e="" />< />
- The three letters H, E, and T can only be arranged in one specific way (HET).
Arranging Remaining Letters
- After fixing H, E, and T, we have the remaining letters O, N, and S to arrange.
- The arrangement can be calculated as follows:
- We have 3 remaining letters: O, N, S.
- Total arrangements of O, N, S = 3! = 6.
Final Calculation
- Since HET is fixed as a single arrangement, the total arrangements satisfying the conditions are:
- Total valid arrangements = Number of arrangements of (O, N, S) × 1
- Total valid arrangements = 6 × 1 = 6.
Conclusion
- The total number of arrangements of the letters in the word "honest" where H precedes E and E precedes T is 120.