Circle x² + y² – 2x – 4y + 4 = 0
$\Rightarrow$ (x – 1)² + (y – 2)² = 1
Centre: (1, 2), radius = 1
Line 3x + 4y – k = 0 intersects the circle at two distinct points.
$\Rightarrow$ distance of centre from the line < radius
$\Rightarrow$ $\left| {{{3 \times 1 + 4 \times 2 - k} \over {\sqrt {{3^2} + {4^2}} }}} \right| < 1$
$\Rightarrow$ |11 - k| < 5
$\Rightarrow$ 6 < k < 5
$\Rightarrow$ k $\in$ {7, 8, 9, ……15} since k $\in$ I
$\therefore$ Total 9 integral value of k.