Four beads - one red, one green, one blue, one yellow - are threaded on a circular bracelet. Two bracelets are the same if one can be rotated into the other. How many different bracelets are there?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Count the arrangements first, then divide.
2The strategy
Count all the ways to place four different beads in four positions, then work out how many of those arrangements are just rotations of each other. Each bracelet appears once per rotation.
3The full solution
There are 4 x 3 x 2 x 1 = 24 arrangements in a line. A circle of four positions has 4 rotations, and every arrangement is counted once per rotation, so 24 divided by 4 = 6.
Solution
The reliable way
First count arrangements as if the positions were fixed: 4 x 3 x 2 x 1 = 24. Now notice that rotating a bracelet by one, two or three positions gives the same bracelet, so each genuine bracelet has been counted 4 times. Divide: 24 divided by 4 = 6. The transferable idea: overcount deliberately, then divide by exactly how many times each object was counted.
The elegant way
Hold the red bead fixed and arrange the other three around it: 3 x 2 x 1 = 6, with no division needed.
Why this is on the test: Counting up to symmetry is a genuine 5-point idea, and it is the first place students meet a deliberate overcount-and-correct.