Question

The number of integers greater than 7000 using 2, 4, 6, 7, 8 without repetition is ________.

• A. 168
• B. 336
• C. 196
• D. 256
• E. 512

Hint:

Don't forget to count numbers with more digits than the lower bound.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
We need to count the total number of integers that can be formed using the digits \{2, 4, 6, 7, 8\} without repetition, which are strictly greater than 7000. Since we have 5 digits available, the integers formed can be either 4-digit numbers or 5-digit numbers. We must consider both cases.
Step 2: Key Formula or Approach
We will use the fundamental principle of counting (the multiplication rule) and consider the cases for 4-digit and 5-digit numbers separately. The total count will be the sum of the counts from each case.
Step 3: Detailed Explanation
Case 1: 5-digit numbers
Any 5-digit number formed using the given digits \{2, 4, 6, 7, 8\} will be greater than 7000 (the smallest would be 24678).
We need to find the number of ways to arrange these 5 distinct digits in 5 positions. This is a permutation of 5 items.
Number of 5-digit numbers = \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
Case 2: 4-digit numbers
To form a 4-digit number greater than 7000, the first digit (the thousands place) must be either 7 or 8.
Let's consider the number of choices for each of the four positions: \(\underline{\hspace{0.5cm}} \ \underline{\hspace{0.5cm}} \ \underline{\hspace{0.5cm}} \ \underline{\hspace{0.5cm}}\)
- Thousands place: It must be 7 or 8. So, there are 2 choices.
- Once the first digit is chosen, we have 4 digits remaining from the original set of 5.
- Hundreds place: We can choose any of the remaining 4 digits.
- Tens place: We can choose any of the remaining 3 digits.
- Units place: We can choose any of the remaining 2 digits.
Total number of 4-digit numbers = \(2 \times 4 \times 3 \times 2 = 48\).
Total Count
The total number of integers greater than 7000 is the sum of the counts from both cases.
Total integers = (Number of 5-digit numbers) + (Number of 4-digit numbers>7000)
Total integers = \(120 + 48 = 168\).
Step 4: Final Answer
The total number of such integers is 168.