Digits available

We can form 5-digit numbers using the digits $\{0,1,2,3,4,5,6,7\}$ Repetition is not mentioned, so we take no repetition.

Let the 5-digit number be $abcde$ where:

• $a$ = first digit
• $e$ = last digit

Condition for being greater than $50000$

Since the number must be greater than $50000$, the first digit must be one of $a\in\{5,6,7\}.$

Condition on first and last digits

We need $a+e\le 8.$

Also, since digits do not repeat, $e\ne a$.

Count possible $(a,e)$ pairs

We check each possible first digit.

Case 1: $a=5$

Then $5+e\le 8 \implies e\le 3.$ Possible last digits: $e\in\{0,1,2,3\}$ So, number of choices = $4$.

Case 2: $a=6$

Then $6+e\le 8 \implies e\le 2.$ Possible last digits: $e\in\{0,1,2\}$ So, number of choices = $3$.

Case 3: $a=7$

Then $7+e\le 8 \implies e\le 1.$ Possible last digits: $e\in\{0,1\}$ So, number of choices = $2$.

Hence total valid choices for first and last digits: $4+3+2=9.$

Fill the middle three places

After fixing $a$ and $e$, we must fill the 2nd, 3rd, and 4th places using the remaining digits.

From 8 digits, 2 are already used, so $6$ digits remain.

Number of ways to arrange 3 distinct digits in the middle is ${}^6P_3 = 6\times 5\times 4 = 120.$

Total count

Therefore, total numbers are $9\times 120 = 1080.$

Check options

The obtained answer is $1080,$ which is not among the given options.

Comparison with stored answer

Stored correct answer is D: 4607, but our correct counting gives $1080.$

So the stored answer appears to be incorrect.