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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
avatar+12725 
+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
avatar+1165 
0

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
avatar+12725 
+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
avatar+1165 
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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
avatar+12725 
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
avatar+1165 
0

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.laugh

GingerAle  Feb 18, 2018
edited by GingerAle  Feb 18, 2018
 #9
avatar+12725 
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
avatar+3023 
+2

Thanks, EP and GingerAle! smiley

tertre  Feb 18, 2018
 #5
avatar+12725 
+1

You're welcome!  Hope we didn't confuse you as much as I often am.  cheeky

ElectricPavlov  Feb 18, 2018

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