Analysis:
To find the value of \(k\) for which the system of equations has no solution, we need to determine the conditions under which the two equations are inconsistent, i.e., they do not have a common solution.

Given Equations:
The given system of equations is:
1. \(x + 2y = 5\)
2. \(3x + ky + 15 = 0\)

Conditions for No Solution:
For the system of equations to have no solution, the two equations must represent parallel lines. This means that the slopes of the two equations must be equal, but their y-intercepts must be different.

Calculating Slopes:
The first equation can be rewritten in the form \(y = mx + c\) as:
\[2y = -x + 5\]
\[y = -\frac{1}{2}x + \frac{5}{2}\]
The slope of the first equation is \(-\frac{1}{2}\).
The second equation can be rewritten as:
\[ky = -3x - 15\]
\[y = -\frac{3}{k}x - \frac{15}{k}\]
The slope of the second equation is \(-\frac{3}{k}\).

Setting Equal Slopes:
For no solution, the slopes of the two equations must be equal:
\[-\frac{1}{2} = -\frac{3}{k}\]
Solving for \(k\):
\[k = 6\]

Conclusion:
Therefore, the value of \(k\) for which the system of equations \(x + 2y = 5\) and \(3x + ky + 15 = 0\) has no solution is \(k = 6\).