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Printjmc
counting and probability junior
Problem
A game board consists of squares that alternate in color between black and white. The figure below shows square in the bottom row and square in the top row. A marker is placed at A step consists of moving the marker onto one of the adjoining white squares in the row above. How many -step paths are there from to (The figure shows a sample path.) 
(A)
(B)
(C)
(D)
Solution
Notice that, in order to step onto any particular white square, the marker must have come from one of the or white squares immediately beneath it (since the marker can only move on white squares). This means that the number of ways to move from to that square is the sum of the numbers of ways to move from to each of the white squares immediately beneath it(also called the Water Fall Method). To solve the problem, we can accordingly construct the following diagram, where each number in a square is calculated as the sum of the numbers on the white squares immediately beneath that square (and thus will represent the number of ways to remove from to that square, as already stated). The answer is therefore .
Final answer
A