How many integers greater than 5000 can be formed with the digit 7, 6, 5, 4 and 3, using each digit at most once?

Options

A. 72
B. 144
C. 84
D. 192

Answer

To solve the problem of how many integers greater than 5000 can be formed using the digits 7, 6, 5, 4, and 3, with each digit used at most once, we can break it down into two cases: forming 4-digit numbers and forming 5-digit numbers.

### Step-by-Step Solution:

**Step 1: Determine the conditions for 4-digit numbers.**
- A 4-digit number must be greater than 5000. Therefore, the first digit (thousands place) must be either 5, 6, or 7.

**Step 2: Count valid 4-digit numbers starting with 5.**
- If the first digit is 5, the remaining digits can be 3, 4, 6, or 7.
- We have 4 options for the second digit, 3 options for the third digit, and 2 options for the fourth digit.
- The number of combinations is calculated as:
\[ 4 \times 3 \times 2 = 24 \]

**Step 3: Count valid 4-digit numbers starting with 6.**
- If the first digit is 6, the remaining digits can be 3, 4, 5, or 7.
- Similarly, we have:
\[ 4 \times 3 \times 2 = 24 \]

**Step 4: Count valid 4-digit numbers starting with 7.**
- If the first digit is 7, the remaining digits can be 3, 4, 5, or 6.
- Again, we have:
\[ 4 \times 3 \times 2 = 24 \]

**Step 5: Total valid 4-digit numbers.**
- Adding all the valid combinations:
\[ 24 + 24 + 24 = 72 \]

**Step 6: Determine the conditions for 5-digit numbers.**
- A 5-digit number can be formed with any of the digits since all 5 digits are greater than 5000.
- The total number of 5-digit combinations is simply the permutations of the 5 digits.

**Step 7: Calculate the number of 5-digit combinations.**
- The number of ways to arrange 5 digits is:
\[ 5! = 120 \]

**Step 8: Combine the results from both cases.**
- The total number of integers greater than 5000 is:
\[ 72 \text{ (from 4-digit numbers)} + 120 \text{ (from 5-digit numbers)} = 192 \]

### Final Answer:
The total number of integers greater than 5000 that can be formed using the digits 7, 6, 5, 4, and 3 is **192**.

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