To determine how many solutions the equation \( 10r - 6r - 16 = 15 + 4r \) has, we can simplify and solve it.
First, simplify the left side:
\[ 10r - 6r = 4r \]
So the equation becomes:
\[ 4r - 16 = 15 + 4r \]
Next, let's subtract \( 4r \) from both sides:
\[ 4r - 4r - 16 = 15 + 4r - 4r \]
This simplifies to:
\[ -16 = 15 \]
The equation \(-16 = 15\) is not true and therefore, there are no values of \( r \) that can satisfy this equation.
Thus, the equation has no solutions.