Let the nine different letters ` A, B, C… I in {1, 2, 3, …, 9}`. Then find the probability that product `(A - 1)(B - 1) … (I - 9)` is an even number.

Explanation

To find the probability that the product \((A - 1)(B - 1) \ldots (I - 9)\) is an even number, we can follow these steps:

### Step 1: Identify the Values of \(A, B, C, \ldots, I\)
The letters \(A, B, C, \ldots, I\) represent the numbers from the set \(\{1, 2, 3, \ldots, 9\}\). Therefore, each letter can take a unique value from this set.

### Step 2: Analyze the Expression \((A - 1)(B - 1) \ldots (I - 9)\)
The expression consists of the terms \((A - 1), (B - 1), \ldots, (I - 9)\). We can rewrite these terms:
- \(A - 1\) can take values from \(0\) to \(8\) (since \(A\) ranges from \(1\) to \(9\)).
- \(B - 1\) can also take values from \(0\) to \(8\).
- Continuing this for all letters, we see that each term will be one less than the corresponding number.

### Step 3: Determine the Parity of Each Term
The values of \(A, B, C, \ldots, I\) can be either odd or even:
- Odd numbers in the set \(\{1, 2, 3, \ldots, 9\}\): \(1, 3, 5, 7, 9\) (5 odd numbers)
- Even numbers in the set: \(2, 4, 6, 8\) (4 even numbers)

When we subtract \(1\) from these:
- Odd numbers become even: \(0, 2, 4, 6, 8\)
- Even numbers become odd: \(1, 3, 5, 7\)

### Step 4: Count the Even and Odd Factors
From the analysis:
- There are \(5\) terms that will be even (from odd numbers) and \(4\) terms that will be odd (from even numbers).

### Step 5: Determine the Conditions for the Product to be Even
For the product \((A - 1)(B - 1) \ldots (I - 9)\) to be even, at least one of the factors must be even. Since we have \(5\) even factors available, it is guaranteed that at least one of the factors will be even.

### Step 6: Calculate the Probability
Since it is certain that the product will always contain at least one even factor, the probability that the product is even is:
\[ P(\text{Product is even}) = 1 \]

### Conclusion
Thus, the required probability that the product \((A - 1)(B - 1) \ldots (I - 9)\) is an even number is:
\[ \text{Probability} = 1 \]

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