Consider the grid of points shown here. Suppose that,...

Consider the grid of points shown here. Suppose that, starting at the point labeled $A,$ you can go one step up or one step to the right at each move. This procedure is continued until the point labeled $B$ is reached. How many different paths from $A$ to $B$ arc possible? Hint: Note that to reach $B$ from $A$, you must take 4 steps to the right and 3 steps upward.

Key Concepts

Lattice Paths

This concept involves moving on a grid (or lattice) following a set of allowed moves—in this case, steps upward or to the right. The paths are typically counted by considering movement along the rows and columns in a grid, and problems often include constraints such as starting and ending points, or allowed directions.

Combinatorial Counting

This refers to the process of counting the number of ways an event can occur under certain conditions. In problems involving grid paths, each valid path can be formulated as an ordering of a fixed number of discrete moves, making it a common application of combinatorial principles.

Binomial Coefficient

The binomial coefficient arises when counting the number of ways to choose positions for one type of move out of the total moves. In path counting on a grid, if a path consists of two types of moves (such as upward and rightward steps), the total number of distinct sequences is given by a binomial coefficient, representing the number of combinations of these moves.

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