Geometric Counting: all problems

How many triangles can be formed by connecting 3 points in the figure below?

How many squares of all sizes can be formed from a 3x3 grid of unit squares?

How many rectangles of any size are in a $4 \times 6$ grid of squares?

A mouse is standing on a grid at location (0,0). There is cheese on the grid at (5,5). He can only move right or up. How many different paths can he take to the cheese?

Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?

How many triangles are in this figure?

Joe is standing at the point (0,0) in the infinite coordinate plane. On any given move, there is a $\frac{1}{3}$ chance of him moving up, $\frac{1}{3}$ chance of him moving right, $\frac{1}{6}$ chance of him moving down, and $\frac{1}{6}$ chance of him moving left. If the probability he will reach the point (3,3) in less than or equal to 8 moves can be described as $\frac{a}{3^b}$, then what is the value of a + b?

How many rectangles of any size are in a $4 \times 6$ grid of squares?

How many triangles can be formed by connecting 3 points in the figure below?

A frog is located at the origin. It makes four hops, each of which moves it either 1 unit to the right or 1 unit to the left. If it also ends at the origin, how many 4-hop paths can it take?

Josh starts at the point (0,0) and needs to travel to his home at the point (6,7). There is a fire at the point (2,3) so he cannot travel there. How many ways are there for Josh to reach his home if he can only move up or right?

Josh starts at the point (0,0) and needs to travel to his home at the point (6,7). There is a fire at the point (2,3) so he cannot travel there. Also, he needs to pick up a book from the library at the point (5,5). How many ways are there for Josh to pick up the book and reach his home if he can only move up or right?

Josh is now home at the point (6,7) but realizes he forgot something at school which is at the point (8,11). Because he is in a rush to get there before it closes, he is now willing to move diagonally (up-right) in addition to being able to go up and right. How many ways can he reach his school?

Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?

How many squares are in a 5x5 grid of squares with 2 diagonally opposite corners missing?

How many rectangles are in the figure below?

Leanne and Jing Jing are walking around the $xy$ plane. In one step, Leanne can move from any point $(x, y)$ to $(x + 1, y)$ or $(x, y + 1)$ and Jing Jing can move from $(x, y)$ to $(x - 2, y + 5)$ or $(x + 3, y - 1)$. The number of ways that Leanne can move from $(0, 0)$ to $(20, 20)$ is equal to the number of ways that Jing Jing can move from $(0, 0)$ to $(a, b)$, where a and b are positive integers. Compute the minimum possible value of $a + b$.