2020 AMC 8 Problems/Problem 21
Contents
[hide]Problem 21
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.)
Solution 1
Noticing that we can only move along white squares, to get to a white square we can only go from the one or two white squares immediately beneath it. In the following diagram, each number represents the number of ways to move from to that square.
So the answer is
~yofro (Diagram credits to franzliszt)
Solution 2
Suppose we "extend" the chessboard indefinitely to the right:
The total number of paths from
to
(including invalid paths which cross over the red line) is
. We subtract the number of invalid paths that pass through
or
. The number of paths that pass through
is
and the number of paths that pass through
is
. However, we overcounted the invalid paths which pass through both
and
, of which there are 2 paths. Hence, the number of invalid paths is
and the number of valid paths from
to
is
. -scrabbler94
See also
2020 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
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