If none of the digits `3, 5, 7, 8, 9` be repeated, how many different numbers greater than 7000 can be formed with them?

Options

A. 190
B. 192
C. 180
D. 185

Explanation

To solve the problem of how many different numbers greater than 7000 can be formed using the digits 3, 5, 7, 8, and 9 without repetition, we will break it down into two cases: forming four-digit numbers and five-digit numbers.

### Step-by-Step Solution:

**Step 1: Identify the digits available.**
The digits we can use are 3, 5, 7, 8, and 9.

**Step 2: Consider four-digit numbers greater than 7000.**
For a four-digit number to be greater than 7000, the first digit must be either 7, 8, or 9.

- **Case 1: First digit is 7.**
- Remaining digits: 3, 5, 8, 9 (4 options left).
- Second digit can be any of the remaining 4 digits.
- Third digit can be any of the remaining 3 digits.
- Fourth digit can be any of the remaining 2 digits.

The total combinations for this case:
\[ 1 \times 4 \times 3 \times 2 = 24 \]

- **Case 2: First digit is 8.**
- Remaining digits: 3, 5, 7, 9 (4 options left).

The total combinations for this case:
\[ 1 \times 4 \times 3 \times 2 = 24 \]

- **Case 3: First digit is 9.**
- Remaining digits: 3, 5, 7, 8 (4 options left).

The total combinations for this case:
\[ 1 \times 4 \times 3 \times 2 = 24 \]

Adding all cases together for four-digit numbers:
\[ 24 + 24 + 24 = 72 \]

**Step 3: Consider five-digit numbers.**
Any five-digit number formed with the digits 3, 5, 7, 8, and 9 will be greater than 7000 since the smallest five-digit number using these digits starts with 3, which is still greater than 7000.

The total combinations for five-digit numbers:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

**Step 4: Combine the results.**
The total number of different numbers greater than 7000 is the sum of the four-digit and five-digit numbers:
\[ 72 + 120 = 192 \]

### Final Answer:
The total number of different numbers greater than 7000 that can be formed using the digits 3, 5, 7, 8, and 9 without repetition is **192**.

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