On the modified chess board `10 xx 10.` Amit and Suresh two persons which start moving towards and the each other person moving with same constant speed. Amit can move only to the right and upwards along the while Suresh can move only to the left or downwards along the lines of the chess board. The alone of ways in with Amit and Suresh can meet at same point during their trip is

Options

A. A) `""^(20)C_(10)`
B. B)`((11)/(10))((10)/(9))((9)/(8))((8)/(7))((7)/(6))((6)/(5))((5)/(4))((4)/(3))((3)/(2))((2)/(1))`
C. C) `2^(10)((1)/(1))((3)/(2))((5)/(3))((7)/(5))* * * * * * * ((19)/(10))`
D. D) `((2)/(1))((6)/(2))((10)/(3))* * * * * * ((38)/(10))`

Answer

To solve the problem of how many ways Amit and Suresh can meet on a 10x10 chessboard, we can follow these steps:

### Step 1: Understand the Movement
Amit can move only to the right and upwards, while Suresh can move only to the left and downwards. This means that they will start from opposite corners of the chessboard. Amit starts from the bottom-left corner (0,0) and Suresh starts from the top-right corner (10,10).

### Step 2: Determine the Total Steps
To meet at a point on the chessboard, both Amit and Suresh will need to take a total of 20 steps (10 steps horizontally and 10 steps vertically).

### Step 3: Define the Meeting Point
Let’s denote the meeting point as (x, y). For Amit to reach (x, y), he must move right x times and up y times. For Suresh to reach (x, y), he must move left (10 - x) times and down (10 - y) times.

### Step 4: Calculate the Total Steps
The total number of steps taken by both Amit and Suresh to meet at point (x, y) is:
- Amit: x (right) + y (up) = x + y
- Suresh: (10 - x) (left) + (10 - y) (down) = 20 - (x + y)

Thus, the total number of steps taken by both is:
\[ \text{Total Steps} = (x + y) + (20 - (x + y)) = 20 \]

### Step 5: Choose Steps
To meet, Amit must take a total of 10 steps right and 10 steps up, while Suresh must take 10 steps left and 10 steps down. The number of ways to arrange these steps can be calculated using combinations.

### Step 6: Use Combinations Formula
The total number of ways Amit can arrange his 10 right and 10 up moves is given by the combination formula:
\[ \text{Ways} = \binom{20}{10} \]

### Step 7: Final Calculation
Thus, the total number of ways in which Amit and Suresh can meet at the same point during their trip is:
\[ \text{Total Ways} = \binom{20}{10} \]

### Conclusion
The answer to the question is \( \binom{20}{10} \).

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