Grid: Passing Through a Point
Starting at the point \((0,0)\), how many ways are there to get to the point \((8,5)\) if you have to pass through the point \((3,2)\), and each step can only be in a positive direction (i.e., to the right or up)?
Answer
We break the problem into two parts:
From \((0,0)\) to \((3,2)\): This requires 3 right moves and 2 up moves (a total of 5 moves). The number of ways to choose the 2 moves up is given by:
\[ \binom{5}{2} = 10. \]From \((3,2)\) to \((8,5)\): This requires \(8-3=5\) right moves and \(5-2=3\) up moves (a total of 8 moves). The number of ways to choose the 3 moves up is given by:
\[ \binom{8}{3} = 56. \]
Multiplying these together, the total number of paths is
\[
10 \times 56 = 560.
\]