If S = {2, 3, 4, 5, 7, 9}, then the number of different three-digit numbers (with all distinct digits) less than 400 that can be formed from S is ------

Options

A. 20
B. 40
C. 80
D. 120

Answer

To solve the problem of finding the number of different three-digit numbers (with all distinct digits) less than 400 that can be formed from the set \( S = \{2, 3, 4, 5, 7, 9\} \), we can follow these steps:

### Step 1: Determine the possible hundreds place digits
Since we want three-digit numbers less than 400, the hundreds place can only be filled with the digits 2 or 3 from the set \( S \).

- **Possible choices for the hundreds place**: 2 or 3

### Step 2: Case Analysis
We will analyze two cases based on the hundreds place digit.

#### Case 1: Hundreds place is 2
- If the hundreds place is 2, we can choose the tens and units places from the remaining digits in \( S \) which are \( \{3, 4, 5, 7, 9\} \).
- We need to choose 2 digits from these 5 remaining digits.

1. **Choosing the tens place**: We have 5 options (3, 4, 5, 7, 9).
2. **Choosing the units place**: After choosing the tens place, we will have 4 digits left to choose from.

So, the number of combinations for this case is:
\[ 5 \text{ (choices for tens)} \times 4 \text{ (choices for units)} = 20 \]

#### Case 2: Hundreds place is 3
- If the hundreds place is 3, we can choose the tens and units places from the remaining digits in \( S \) which are \( \{2, 4, 5, 7, 9\} \).
- Again, we need to choose 2 digits from these 5 remaining digits.

1. **Choosing the tens place**: We have 5 options (2, 4, 5, 7, 9).
2. **Choosing the units place**: After choosing the tens place, we will have 4 digits left to choose from.

So, the number of combinations for this case is:
\[ 5 \text{ (choices for tens)} \times 4 \text{ (choices for units)} = 20 \]

### Step 3: Total Combinations
Now, we add the results from both cases:
\[ \text{Total} = \text{Case 1} + \text{Case 2} = 20 + 20 = 40 \]

### Final Answer
The number of different three-digit numbers (with all distinct digits) less than 400 that can be formed from \( S \) is **40**.

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