The number of elements in the set { z = a + i b ∈ ℂ : a , b ∈ ℤ and 1 < z - 3 +…

We know that $\left|z-\left(3-2i\right)\right|=r$ represents a circle with centre $\left(3,-2\right)$ and radius $r$.

$1<\left|z-3+2i\right|<4$

$1<{\left(a-3\right)}^2+{\left(b+2\right)}^2<16$

The ordered pairs of $\left(a,b\right)$ satisfying the above inequality are $\left(0,\pm 2\right),\left(\pm 2,0\right),\left(\pm 1,\pm 2\right),\left(\pm 2,\pm 1\right)$

$\left(\pm 2,\pm 3\right),\left(3\pm ,\pm 2\right),\left(\pm 1,\pm 1\right),\left(2\pm ,\pm 2\right)$

$\left(\pm 3,0\right),\left(0,\pm 3\right),\left(\pm 3\pm 1\right),\left(\pm 1,\pm 3\right)$

i.e. total $40$ elements are present in the given set