The standard form of a circle's equation is:
\[
(x - h)^2 + (y - k)^2 = r^2,
\]
where \( (h, k) \) is the center and \( r \) is the radius.
Step 1: Identify given data.
The center is \( (5, 4) \), so \( h = 5 \) and \( k = 4 \). The circle touches the \( Y \)-axis, meaning the radius \( r \) equals the \( x \)-coordinate of the center, so \( r = 5 \).
Step 2: Substitute values into the equation.
Using the center \( (5, 4) \) and radius \( r = 5 \), the equation is:
\[
(x - 5)^2 + (y - 4)^2 = 5^2.
\]
Step 3: Simplify the equation.
Expand the binomials:
\[
(x - 5)^2 = x^2 - 10x + 25, \quad (y - 4)^2 = y^2 - 8y + 16.
\]
Substitute these into the equation:
\[
x^2 - 10x + 25 + y^2 - 8y + 16 = 25.
\]
Simplify:
\[
x^2 + y^2 - 10x - 8y + 16 = 0.
\]
Step 4: Check the corresponding option.
The derived equation matches option \( \mathbf{(2)} \).
Final Answer:
The equation of the circle is:
\[
\boxed{x^2 + y^2 - 10x - 8y + 16 = 0}.
\]