Question

With rectangular axes of coordinates, the number of paths from (1,1) to (8,10) via (4,6), where each step from any point (x, y) is either to (x, y+1) or to (x+1, y), is

• A. 3290
• B. 3920
• C. 3940
• D. 3950

Answer 1

The Correct Option is A

Let the coordinates of points be: 

• A = (1, 1)
• C = (4, 6)
• B = (8, 10)

Step 1: Number of paths from A to C

From A(1, 1) to C(4, 6), the total steps right = \(4 - 1 = 3\), and up = \(6 - 1 = 5\), for a total of \(3 + 5 = 8\) steps.

So, the number of ways to reach C from A is: \[ \binom{8}{5} \]

Step 2: Number of paths from C to B

From C(4, 6) to B(8, 10), the total steps right = \(8 - 4 = 4\), and up = \(10 - 6 = 4\), for a total of \(4 + 4 = 8\) steps.

So, the number of ways to reach B from C is: \[ \binom{8}{4} \]

Step 3: Total number of paths from A to B via C

Multiply the two: \[ \binom{8}{5} \times \binom{8}{4} = 56 \times 70 = \boxed{3920} \]