1. We must form integers greater than $7000$ using the digits $3,5,6,7,8$ without repetition.

2. Since the number must be greater than $7000$, it can be:

   • a $4$-digit number greater than $7000$, or
   • a $5$-digit number.

Case 1: $4$-digit numbers greater than $7000$

For a $4$-digit number to be greater than $7000$, its thousand's digit must be either $7$ or $8$.

(i) Thousand's digit = $7$

The remaining $3$ places can be filled using any $3$ of the remaining $4$ digits $\{3,5,6,8\}$ without repetition.

Number of ways: $^4P_3 = 4 \times 3 \times 2 = 24$

(ii) Thousand's digit = $8$

The remaining $3$ places can be filled using any $3$ of the remaining $4$ digits $\{3,5,6,7\}$ without repetition.

Number of ways: $^4P_3 = 24$

So total $4$-digit numbers: $24+24=48$

Case 2: $5$-digit numbers

Any $5$-digit number formed using all $5$ digits is automatically greater than $7000$.

Number of such numbers: $5! = 120$

Total numbers

$48+120=168$

So the required number of integers is $\boxed{168}$

Checking options

• A: $48$ ❌
• B: $120$ ❌
• C: $168$ ✅
• D: $220$ ❌

Therefore, the correct option is C.