+355 Find the number of ways of choosing three circles below, so that no two circles are next to each other.
There are 12 circles, in the shape of a circle.
[asy]
unitsize(1 cm);
int i;
for (i = 0; i <= 11; ++i) {
draw(Circle(2*dir(360/12*i),0.3));
label(string(i + 1), 2*dir(90 - 360/12*i));
}
[/asy]
+1911 We can solve this problem by considering the empty spaces between the circles. There are 11 spaces between the circles, since there are 12 circles forming a circle.
Since no two chosen circles can be next to each other, the chosen circles must correspond to every other empty space. There are C(11,3) = 165 ways to choose 3 empty spaces out of 11.
