In how many ways can 4 distinct beads be placed on a bracelet that has a knot in it? (Beads cannot slide past the knot. Two ways are considered the same if one is a reflection of the other, since we can flip the bracelet over.)
Similar to how many ways can ten people line up while waitng for a bus: 10!
How many ways can 4 people (beads) line up on a bracelet? 4! = 24 BUT 1/2 of these will simply be 'reflections' and are disallowed.....so 24 x 1/2 = 12 ways. (my guess....anyone else?)
Unless the people line up in a circle, it’s not the same.
For a bracelet without a knot, there are (4-1)! / 2 arrangements. One (1) is subtracted to fix the position of a bead, because the beads can be rotated such that one particular bead can be placed in a start position without rearranging them. The division by two (2) is because the bracelet can be flipped.
A bracelet with a knot prevents rotation. Treat the knot as an additional bead, fixed in position, giving (5-1)! / 2 = 12 arrangements.
Hmmmmm...... the question stated there is a knot (bus door essentially) so the beads CANNOT rotate......so I got 12 too.
Your answer matches because of coincidence, not because of logical analyses.
You said the passengers for the bus are standing in a line, that’s not a circle. If they were standing in a circle, and the bus door prevented them from rotating, then your solution is wrong because you have to count the bus door as a passenger, and you didn’t do that.
Your methodology solves a string of beads in a line, not attached at the end, or a bracelet (ends attached) and each rotation counting as a unique arrangement –this isn’t normally done in circle combinations.
GA.....I'm still confused I guess. The knot in the bracelet makes it essentially 'knot' (pun) a circle but a straight line configuration......the beads cannot rotate...they are in a straight line. So I (think) my naswer is correct, not coincidental. Maybe we are viewing the question differently in our minds' eyes?
You’re right! I withdraw my critical assessment.
The knot does equate to a straight line; changing from the circular permutation P=(n-1) to the linear P=n!
The caveat for reflections (/2) being equivalent for a linear permutation seems unusual, but there isn’t a reason you couldn’t do it. When I scratch my left ear while looking in a mirror I see me scratching my right ear in Alice’s universe, but were both content.