I have a circular necklace with $18$ beads on it. All the beads are different. Making two cuts with a pair of scissors, I can divide the necklace into two strings of beads. If I want each string to have at least $6$ beads, how many different pairs of strings can I make?

Solution:

There are four general types we can make: (6,12), (7,11), (8,10), (9,9).

First type: (6,12), there are 18 possible ways to choose those 6, which are going to be cut from the necklace: choose a direction to count the beads, and choose a starting position (between 2 beads). There are exactly 18 starting positions, since there are 18 spaces between the beads. One cut is equivalent for a pair we can make.

Second type: (7,11), with the same reasoning, there are 18 possible ways to cut the necklace.

Third type: (8,10), with the same reasoning, there are 18 possible ways to cut the necklace.

Fourth type: (9,9), The same reasoning cannot be applied again, since half of the cuts would be exactly the same as the other half. So there are 9 possible cuts, exactly one or each axis of symmetry for the necklace.

The solution will be the sum these values: 18+18+18+9=63