Answer:
192 total numbers
78 even numbers
Step-by-step explanation:
To form a number greater than 5000, it can either have 5 digits, or 4 digits and start with at least a 5.
For the 5 digit case, there are 5! = 120 numbers.
For the 4 digit case, the first digit must be either 5, 6, or 7, giving 3 possibilities. The last three digits can be anything, so there are 4∗3∗2=24 possibilities there. In total, that gives 72 possible 4-digit numbers.
Therefore, there are 192 total numbers greater than 5000 that can be formed with those digits.
With the condition that these must be even, we must still consider similar cases.
For the case with 5 digits, there are 2 choices for the last digit, and 4! = 24 choices for the first 4. Therefore, there are 48 total 5 digit even numbers.
For the case with 4 digit numbers, there are 2 subcases we must consider: with 5 or 7 as the first digit, or with 6 as the first digit.
When the first digit is either 5 or 7, there are 2 choices for the first digit, 2 choices for the last digit, and 6 ways to choose the middle. There are therefore 24 possibilities in this case.
When the first digit is 6, there is one possibility for the first digit and one possibility for the last digit, along with the 6 choices in the middle. Therefore there are 6 numbers in this case, and 30 4-digit evens in all.
Adding these up, there are 78 even numbers greater than 5000 that can be made with these digits.
