To solve the problem of determining the number of integers greater than 7000 that can be formed using the digits 3, 5, 6, 7, and 8 without repetition, follow these steps:
We begin by noting that a valid integer greater than 7000 must be a 4-digit or 5-digit number.
Firstly, consider forming 4-digit numbers from the given digits: 3, 5, 6, 7, and 8. The leading digit must be greater than 7 to ensure the number is greater than 7000. Therefore, the leading digit can be 8.
With 8 as the leading digit, we are left with the digits: 3, 5, 6, and 7. We need to choose 3 more digits from these remaining four digits to form a 4-digit number.
The number of ways to choose 3 digits from 4 is given by the permutation:
P(4,3), which is equal to 4!/(4-3)! = 4! = 4 × 3 × 2 = 24 ways.
Next, consider forming 5-digit numbers from all five given digits: 3, 5, 6, 7, and 8. We can arrange these 5 digits in 5! ways since all digits must be used, leading to:
5! = 5 × 4 × 3 × 2 × 1 = 120 ways.
Add the results from both cases to find the total number of integers greater than 7000 that can be formed:
4-digit numbers: 24
5-digit numbers: 120
Thus, the total is 24 + 120 = 144 + 24 = 168.
Therefore, the total number of integers greater than 7000 that can be formed with the digits given, without repetition, is 168.
