Management city has m parallel roads running East-West and n parallel roads running North-South. How many shortest possible routes are possible to go from one corner of the city to its diagonally opposite corner?

Options

A. `"^(m+n)C_(m-1)`
B. `"^(m+n)C_n`
C. `"^(m+n-2)C_(m-1)`
D. none of these

Answer

To solve the problem of finding the number of shortest possible routes from one corner of the city to its diagonally opposite corner, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Grid Layout**:
- We have `m` parallel roads running East-West and `n` parallel roads running North-South.
- To move from one corner to the diagonally opposite corner, we need to move `m-1` units in the North-South direction and `n-1` units in the East-West direction.

2. **Total Moves Calculation**:
- The total number of moves required to reach the opposite corner is the sum of the moves in both directions:
\[ \text{Total Moves} = (m - 1) + (n - 1) = m + n - 2 \]

3. **Choosing Moves**:
- Out of the total moves, we need to choose `m-1` moves to go North (or equivalently `n-1` moves to go East).
- The number of ways to arrange these moves can be calculated using combinations.

4. **Using Combinations**:
- The number of ways to choose `m-1` moves from a total of `m+n-2` moves is given by the combination formula:
\[ \text{Number of Routes} = \binom{m+n-2}{m-1} \quad \text{or} \quad \binom{m+n-2}{n-1} \]
- Both expressions are equivalent because choosing `m-1` North moves automatically determines the `n-1` East moves and vice versa.

5. **Final Answer**:
- Therefore, the total number of shortest possible routes from one corner of the city to its diagonally opposite corner is:
\[ \text{Number of Routes} = \binom{m+n-2}{m-1} \]