We know that the generic equation of a circle is:
\((x - h)^{2} + (y - k)^{2} = r^{2}\)

where $(h, k)$ is the center of the circle and r is the radius of the circle.

We are given the equation:
\(x^{2} + y^{2} = 16\)

\((x - 0)^{2} + (y - 0)^{2} = 4^{2}\)

By taking a look at it, we can see that the value of (h, k) is (0, 0) here. Which means that the center of the circle is on the origin (0, 0). Also, the radius of this circle is 4.

The figure of this circle is given as follows:


We can either check the figure to find out the total number of integer coordinates inside the circle or exactly on the circle. Or we can also check this by putting different values of x and y and find out the answer as well.

If we put different positive and negative integer values of x and y, then the answer of left hand side of the equation has to be less than or exactly equal to 16. Any coordinate that produces a result greater than 16 is not our answer as it will be outside the circle.

\(x^{2} + y^{2} = 16\)

- We can see that if we put $x = 0$, then we can put y = 1, 2, 3, 4 in the above equation and our result will be less than or equal to 16. Similarly, we can put y = -1, -2, -3, -4 as well. So these are 8 possible coordinates already.
- We can also put x = 1, 2, 3, 4 or x = -1, -2, -3, -4 for y = 0, which means these are 8 more coordinates.
- We can put a combination of x = 1 and y = 1, 2, 3. Similarly x = 2 and y = 1, 2, 3. Also, x = 3 and y = 1, 2. These are 8 combinations for one quadrant. There are four such quadrants. So we can multiply 8 * 4 = 32 coordinates.
- Lastly, (0, 0) coordinate also lies inside the circle.


So in total there are 8 + 8 + 32 + 1 = 49 coordinates inside the circle.

Hence, the correct answer is 49.
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solution please.

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