A number of 4 different digits is formed by using the digits 1, 2, 3, 4, 5, 6, 7 in all possible ways without repetition.
How many of them are greater than 3400?

Options

A. 840
B. 560
C. 480
D. 120

Answer

To solve the problem of how many four-digit numbers greater than 3400 can be formed using the digits 1, 2, 3, 4, 5, 6, and 7 without repetition, we can break down the solution into several steps.

### Step-by-Step Solution:

1. **Identify the Range of the First Digit**:
- Since we want the number to be greater than 3400, the first digit can only be 4, 5, 6, or 7 (it cannot be 1, 2, or 3).
- Therefore, we have 4 options for the first digit: {4, 5, 6, 7}.

2. **Case Analysis Based on the First Digit**:
- We will analyze each case based on the choice of the first digit.

**Case 1: First digit is 4**
- The number must be greater than 3400, so the second digit can be 1, 2, 3, 5, 6, or 7. Thus, we have 6 options.
- After choosing the first and second digits, we have 5 remaining digits for the third digit and 4 remaining digits for the last digit.
- Total combinations for this case:
\[ 6 \times 5 \times 4 = 120 \]

**Case 2: First digit is 5**
- The second digit can be any of the remaining 6 digits (1, 2, 3, 4, 6, 7).
- Total combinations for this case:
\[ 6 \times 5 \times 4 = 120 \]

**Case 3: First digit is 6**
- The second digit can be any of the remaining 6 digits (1, 2, 3, 4, 5, 7).
- Total combinations for this case:
\[ 6 \times 5 \times 4 = 120 \]

**Case 4: First digit is 7**
- The second digit can be any of the remaining 6 digits (1, 2, 3, 4, 5, 6).
- Total combinations for this case:
\[ 6 \times 5 \times 4 = 120 \]

3. **Summing All Cases**:
- Now, we sum the total combinations from all cases:
\[ 120 (from \, 4) + 120 (from \, 5) + 120 (from \, 6) + 120 (from \, 7) = 480 \]

### Final Answer:
The total number of four-digit numbers greater than 3400 that can be formed using the digits 1, 2, 3, 4, 5, 6, and 7 without repetition is **480**.