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First, place the 3 P’s. This creates 4 possible gaps (before, between, after) where H’s can go:
\_ P \_ P \_ P \_ → 4 gaps
We need to place 4 H’s into these 4 gaps, with **at most one H per gap** to prevent adjacency. But we have 4 H’s and only 4 gaps, so we must place exactly one H in each gap.
There is only **one** way to choose positions: H in each gap → so the sequence is:
H P H P H P H → i.e., H P H P H P H
But wait — this places H in all 4 gaps, but total positions:
Each gap holding one H gives:
positions: H (1), P (2), H (3), P (4), H (5), P (6), H (7) → exactly 7 positions.
So the structure is uniquely determined: H in positions 1,3,5,7 and P in 2,4,6.
No other configuration allows 4 non-adjacent H’s in 7 positions.
