A, B, C are the centers of the three circles, each of which touches the other. The radius of circle with center C is 10 units.

Extend the line CA to make the radius of the circle C as CD , and extend the line CB to make another radius of the circle C as CE , Please see image attached.

So we know the line CD is the radius of the circle C i.e. CD = 10 , thus if we consider the value of DA =r1 (which is same as radius of the Circle A) , then CA=10-r1
Similarly we know the line CE is the radius of the circle C i.e. CE = 10, thus if we consider the value of BE = r2 (which is same as the radius of the circle B), then CB = 10-r2
now we know the point A and B are the centers of circle A and B , thus the line segment AB = r1 + r2
Thus perimeter of triangle ABC = CA+AB+CB = 10-r1 + r1+r2 + 10-r2 = 20
answer C .

PS. Sorry for my bad drawing