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.)

tertre
Feb 17, 2018

#1**+1 **

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?)

ElectricPavlov
Feb 18, 2018

#2**+1 **

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.

GA

GingerAle
Feb 18, 2018

#3**+1 **

Hmmmmm...... the question stated there is a knot (bus door essentially) so the beads CANNOT rotate......so I got 12 too.

ElectricPavlov
Feb 18, 2018

#6**0 **

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.

GingerAle
Feb 18, 2018

#7**0 **

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?

ElectricPavlov
Feb 18, 2018

#8**+1 **

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.

GingerAle
Feb 18, 2018

#9**0 **

Thanx for your help GA ! I am still getting 're-acquainted' with this stuff.......can you backcheck my answer for the 5 Repubs and 5 Dems sitting around a table??? G'Day ... ~EP

ElectricPavlov
Feb 18, 2018

#4