The circles x^2+y^2=4 and (x-3)^2+y^2=1 have exactly how many common tangents?

The centres are $(0,0)$ and $(3,0)$ with radii $2$ and $1$, so the distance between the centres is $3=2+1$, the sum of the radii. The circles touch externally, which gives $2$ direct common tangents and $1$ common tangent at the point of contact, a total of $3$. Two circles apart would have $4$, intersecting circles $2$, and internally touching circles $1$. The relation between the centre distance and the radii decides the count.