Three questions on one idea: a thing you count on a grid is a CHOICE, not a picture to draw. A route is a choice of which steps go right, a triangle a choice of three dots, a rectangle a choice of two columns and two rows.
The answer key prints on a separate page.
Section 1 walks a grid of streets from one corner to the far one, only ever going right or up, and counts the routes. Every route is the same length — so many steps right and so many up — so a route is nothing but a choice of which of those steps are the rights, a single combination. The trap is thinking each corner is a fresh right-or-up choice, which would give two to a power and count routes that run clean off the grid.
Section 2 marks a grid of dots and counts the triangles with three dots as corners. Choosing three of the dots is one combination, C(number of dots, 3) — but three dots on a straight line make no triangle, so the rows, the columns and the slanted lines of three come back off. A 3-by-3 grid gives 84 ways to choose three dots and only 76 triangles; miss the two diagonals and you get 78. Section 3 counts the rectangles in the same kind of grid, and it is a different combination: a rectangle needs two of the columns for its left and right edges and two of the rows for its top and bottom, so the count is one combination times another. The tempting wrong answer is the number of little unit cells, which forgets every larger rectangle.
