On the normal chess board as shown, `I_1, & I_2`, are two insects whichstarts moving towards each other. Each insect moving with thesame constant speed. Insect `I_1`, can move only to the right or upwardalong the lines while the insect `I_2`, can move only to the left or downwardalong the lines of the chess board. Find the total number of ways thetwo insects can meet at same point during their trip.

Options

A. A. `((9)/(8))((10)/(7))((11)/(6))((12)/(5))((13)/(4))((14)/(3))((15)/(2))((16)/(1))`
B. B. `2^(8)((1)/(1))((3)/(2))((5)/(3))((7)/(4))((9)/(5))((11)/(6))((13)/(7))((15)/(8))`
C. C. `((2)/(1))((6)/(2))((10)/(3))((14)/(4))((18)/(5))((22)/(6))((26)/(7))((30)/(8))`
D. D. `""^(16)C_(8)`

Answer

To solve the problem of how many ways the two insects can meet on a chessboard, we can follow these steps:

### Step 1: Understand the movement of the insects
Insect \( I_1 \) can move only to the right or upward, while insect \( I_2 \) can move only to the left or downward. They start at opposite corners of the chessboard.

### Step 2: Set the positions
Assume insect \( I_1 \) starts at the bottom-left corner (0,0) and insect \( I_2 \) starts at the top-right corner (8,8) of an 8x8 chessboard.

### Step 3: Determine the total distance
Both insects need to travel across the chessboard. The total distance each insect needs to cover is 8 units horizontally and 8 units vertically, making a total of 16 moves.

### Step 4: Calculate the number of steps
The insects can meet at any point on the chessboard. However, the minimum number of steps for both insects to meet is 8 steps. This is because both insects must take 8 steps in total to meet at a point on the chessboard.

### Step 5: Use combinations to find the number of ways
The total number of ways the two insects can meet is given by the combination formula \( C(n, r) \), where \( n \) is the total number of steps and \( r \) is the number of steps taken in one direction.

Since both insects together take 16 steps (8 right/up for \( I_1 \) and 8 left/down for \( I_2 \)), we need to choose 8 steps from these 16 total steps. Thus, the number of ways they can meet is given by:

\[ \text{Number of ways} = C(16, 8) \]

### Step 6: Final calculation
Using the combination formula:

\[ C(16, 8) = \frac{16!}{8! \cdot 8!} \]

This gives us the total number of ways the two insects can meet at the same point on the chessboard.

### Conclusion
Thus, the total number of ways the two insects can meet at the same point during their trip is \( C(16, 8) \).

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