A black ant has to go from the origin to a point (6, 4) on the Cartesian plane, using the coordinate axes. However, it avoids crossing the point (4, 1), as a red ant is sitting there. Find the total number of shortest paths that it can go along.

Options

A. (A) 150
B. (B) 160
C. (C) 190
D. (D) 240

Answer

To solve the problem of finding the total number of shortest paths from the origin (0, 0) to the point (6, 4) while avoiding the point (4, 1), we can break it down into several steps.

### Step 1: Calculate the total number of shortest paths from (0, 0) to (6, 4)

To get from (0, 0) to (6, 4), the black ant needs to make a total of 6 moves to the right (R) and 4 moves up (U). The total number of moves is 6 + 4 = 10 moves.

The number of ways to arrange these moves can be calculated using the formula for combinations:

\[ \text{Total paths} = \binom{10}{6} = \frac{10!}{6!4!} \]

Calculating this gives:

\[ \binom{10}{6} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \]

### Step 2: Calculate the number of paths that pass through (4, 1)

Next, we need to find the number of paths from (0, 0) to (4, 1) and then from (4, 1) to (6, 4).

**From (0, 0) to (4, 1):**

To go from (0, 0) to (4, 1), the ant must make 4 moves to the right and 1 move up. The total number of moves is 4 + 1 = 5 moves.

The number of ways to arrange these moves is:

\[ \text{Paths to (4, 1)} = \binom{5}{4} = \frac{5!}{4!1!} = 5 \]

**From (4, 1) to (6, 4):**

From (4, 1) to (6, 4), the ant needs to make 2 moves to the right and 3 moves up. The total number of moves is 2 + 3 = 5 moves.

The number of ways to arrange these moves is:

\[ \text{Paths from (4, 1) to (6, 4)} = \binom{5}{2} = \frac{5!}{2!3!} = 10 \]

### Step 3: Calculate the total number of paths passing through (4, 1)

Now we can find the total number of paths that pass through (4, 1) by multiplying the number of paths to (4, 1) and the number of paths from (4, 1) to (6, 4):

\[ \text{Total paths through (4, 1)} = 5 \times 10 = 50 \]

### Step 4: Subtract the paths through (4, 1) from the total paths

Finally, to find the number of paths that do not pass through (4, 1), we subtract the number of paths that pass through (4, 1) from the total number of paths:

\[ \text{Required paths} = \text{Total paths} - \text{Paths through (4, 1)} = 210 - 50 = 160 \]

Thus, the total number of shortest paths that the black ant can take from (0, 0) to (6, 4) while avoiding (4, 1) is **160**.