A coin with heads facing up is shown as 
and a coin with tails facing up is shown as 
.
Six coins are placed in the Starting Arrangement, as shown in the figure below. A “step” is defined as interchanging a pair of adjacent coins without flipping them. The minimum number of steps needed to go from the Starting Arrangement to the Final Arrangement, as shown in the figure, is ________.
The problem requires determining the minimum number of steps needed to rearrange coins from the "Starting Arrangement" to the "Final Arrangement" by interchanging adjacent coins.
Let's define the starting arrangement as \( HHHHTT \) and the final arrangement as \( TTTTHH \).
- The first step is to move the 'H' from the first position over to where the 'T' needs to be placed:
- Interchange first 'H' at position 1 with a 'T' at position 4 → \( HHHTHT \)
- Move 'H' again left to right to position 3:
- Interchange 'H' at position 2 with 'T' at position 3 → \( HHTHHT \)
- Continue moving 'H' left to right:
- Interchange 'H' at position 3 with 'T' at position 2 → \( HTHTHH \)
- Move 'H' from position 2 over to left to complete the arrangement:
- Interchange 'H' at position 1 with 'T' at position 0 → \( THTHHH \)
- The remaining 'T's are lined up on the left. Interchange pairs to correctly align remaining coins for achieving final configuration:
- Interchange ‘H’ from positions 4 to 0 and T from 0 to 4 iteratively – requires a total of 6 more steps:
- \( TTHTHH \) → \( TTHTHH \) → \( TTTTHH \)
- Interchange ‘H’ from positions 4 to 0 and T from 0 to 4 iteratively – requires a total of 6 more steps:
Each step involves only interchanging adjacent coins and counting up leads to a minimum of 9 steps.
Thus, the correct answer is: 9.
Important Questions from Permutations
How many words can be formed with the letters of the word 'POSTMAN', if every word begins with T and ends with M?
In how many ways can cells in a $3 \times 3$ grid be shaded, such that each row and each column have exactly one shaded cell? An example of one valid shading is shown.
- Three husband-wife pairs are to be seated at a circular table that has six identical chairs. Seating arrangements are defined only by the relative position of the people. How many seating arrangements are possible such that every husband sits next to his wife?
- The number of 'three-digit numbers' that can be formed using the digits from 1 to 9 without the repetition of each digit is ________.
- The number of ways in which the letters in the word MINING can be arranged is
