Write down an expression for the number of binomial paths which cross a rectangle from the point (0,0) to the point (3n,2m), with n,m > 1, but which are constrained to have a vertex on the point (n,m)?
To derive an expression for the number of binomial paths that cross a rectangle from (0, 0) to (3n, 2m) with a vertex at (n, m), we can break down the problem into smaller steps.
Step 1: Count the number of paths from (0, 0) to (n, m).
We can use the binomial coefficient to count the number of paths from (0, 0) to (n, m). The binomial coefficient "C(n, k)" counts the number of ways to choose k items from a set of n distinct items. In this case, we can count the number of paths that cross a rectangle by using the formula C(n, m).
Step 2: Count the number of paths from (n, m) to (3n, 2m).
Similarly, we can count the number of paths from (n, m) to (3n, 2m) using the binomial coefficient. In this case, we need to travel a distance of (2n, m), so the number of paths can be counted as C(2n, m).
Step 3: Combine the two steps to get the answer.
To get the complete expression, we multiply the number of paths from (0, 0) to (n, m) by the number of paths from (n, m) to (3n, 2m). This can be written as:
C(n, m) * C(2n, m)
Therefore, the expression for the number of binomial paths that cross a rectangle from (0, 0) to (3n, 2m) with a vertex at (n, m) is C(n, m) * C(2n, m).