A coin with heads facing up is shown as H, and a coin with tails facing up is shown as T.
Six coins are placed in the Starting Arrangement shown 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 is ________.
Figure Q.5 — starting and final coin arrangements
Options:
(A) 3
(B) 6
(C) 9
(D) 12
Official Answer Key:
C
Step-by-Step Solution & Explanation
The minimum number of adjacent transpositions required to convert one arrangement into another equals the number of inversions between them.
In H H H T T T → T T T H H H, every H must end up to the right of every T. There are 3 H’s and 3 T’s, and each H–T pair must cross exactly once.
Minimum steps = 3 × 3 = 9.
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