A game board consists of $64$ squares that alternate between shaded and unshaded. The figure below shows square $P$ in the bottom row and square $Q$ in the top row. A marker is placed at $P.$ A step consists of moving the marker onto one of the adjoining unshaded squares in the row above. How many $7$-step paths are there from $P$ to $Q?$ (The figure shows a sample path.)

Video solution:

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Written solution:

Each move must go up one row and either left or right. Counting row by row from $P,$ each reachable square gets the sum of the counts from the two adjoining squares below it.

The count at $Q$ is $28,$ so there are $28$ paths.

Thus, the correct answer is A.