A wheel is rolled without slipping through $15$ laps on a circular race course with radius $7$. The wheel is perfectly circular and has radius $5$. After the three laps, how many revolutions around its axis has the wheel been turned through?
1. First, calculate the circumference of the circular race course. The formula for the circumference \(C\) of a circle is given by:
   \[
   C = 2 \pi r
   \]
   where \(r\) is the radius of the circle. Here, the radius of the race course is \(7\). Therefore, the circumference of the race course is:
   \[
   C = 2 \pi \times 7 = 14 \pi
   \]

2. Next, determine the total distance the wheel travels after completing \(15\) laps around the race course. This is given by:
   \[
   \text{Total distance} = 15 \times \text{circumference of the race course} = 15 \times 14 \pi = 210 \pi
   \]

3. Now, calculate the circumference of the wheel. The radius of the wheel is \(5\). Therefore, the circumference of the wheel is:
   \[
   \text{Circumference of the wheel} = 2 \pi \times 5 = 10 \pi
   \]

4. Finally, determine the number of revolutions the wheel makes by dividing the total distance traveled by the circumference of the wheel:
   \[
   \text{Number of revolutions} = \frac{\text{Total distance}}{\text{Circumference of the wheel}} = \frac{210 \pi}{10 \pi} = 21
   \]

The final answer is \(\boxed{21}\).