To determine how many solutions the equation \(-15r + 12 = 3(-15r + 4)\) has, we can start by simplifying the equation.
First, distribute the \(3\) on the right side:
\[ 3(-15r + 4) = 3 \cdot -15r + 3 \cdot 4 = -45r + 12 \]
Now, rewrite the original equation with this simplification:
\[ -15r + 12 = -45r + 12 \]
Next, let's isolate the terms involving \(r\). To do this, we can add \(45r\) to both sides:
\[ -15r + 45r + 12 = 12 \]
This simplifies to:
\[ 30r + 12 = 12 \]
Now, we will subtract \(12\) from both sides:
\[ 30r = 0 \]
Next, divide both sides by \(30\):
\[ r = 0 \]
Since we found a specific value for \(r\), this means there is exactly one solution to the equation.
Thus, the number of solutions is \( \boxed{1} \).