Introduction:
We are given three points in a three-dimensional coordinate system: (-4,2,5), (k,1,-2), and (3,1,k). We need to determine the number of values of k for which the point (-4,2,5) lies on the line joining the other two points.

Approach:
To determine if the point (-4,2,5) lies on the line joining the two given points, we can use the concept of collinearity. If the three points are collinear, then the line passing through the first two points will also pass through the third point.

Step 1: Find the direction vector of the line:
To find the direction vector of the line passing through the points (k,1,-2) and (3,1,k), we can subtract the coordinates of the two points.

Direction vector = (3,1,k) - (k,1,-2) = (3-k, 0, k+2)

Step 2: Find the position vector of the point:
The position vector of the point (-4,2,5) is simply (-4,2,5).

Step 3: Determine collinearity:
To check if the three points are collinear, we can compare the direction vector and the position vector. If the position vector is a scalar multiple of the direction vector, then the points are collinear.

If the three points are collinear, we can write the following equation:

(-4,2,5) = t(3-k, 0, k+2)

This equation can be expanded into three separate equations as follows:

-4 = t(3-k)
2 = 0
5 = t(k+2)

Step 4: Solve the equations:
From the second equation, we can see that the value of t is 0, which implies that the points are not collinear. Therefore, there are no values of k for which the point (-4,2,5) lies on the line joining the points (k,1,-2) and (3,1,k).

Conclusion:
After solving the equations, we find that there are no values of k for which the point (-4,2,5) lies on the line joining the points (k,1,-2) and (3,1,k). Therefore, the correct answer is (b) 0.