Paths across a grid
On a grid of cells, a token starts in the top-left cell and must reach the bottom-right cell, moving one cell to the right or one cell down at each step.
How many different paths are there?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
How many moves to the right and how many moves down does every path make?
A path is fixed by choosing which of its moves go right.
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Every path makes the same moves, 5 to the right and 5 down, in some order. Counting paths means counting the orders.
Step 1. From the top-left cell to the bottom-right cell, the token must move 5 columns right and 5 rows down: 10 moves in total.
Step 2. A path is determined by which 5 of the 10 moves go right:
Read carefully. If the question were about a grid of squares and the path ran along the grid lines from one corner to the opposite corner, there would be 6 moves of each kind and paths.