http://www.dworin.com/necklace-slide/
A necklace is made out of identically shaped, coloured beads. Ten beads are red, 9 are black, 7 are white, 12?
A necklace is made out of identically shaped, coloured beads. Ten beads are red, 9 are black, 7 are white, 12 are pink, and 4 are purple. The beads are strung together on a waxed string. The ends are tied together to form a large knot that prevents the beads from sliding past it. How many different necklaces can be made in this way?
Since the knot is there, you get to work with linear permutations (not cyclic ones). So, really, you want to know how many orders you can make with the beads. The tricky part is that, with a necklace, you can flip it over. So if the two orders may actually make the same necklace. The unfortunate part is that this may only be true SOMETIMES… if the necklace is symmetric, then flipping it over doesn’t change it.
The general theorem here is know as “Burnside’s Lemma”. You can check it out here:
http://en.wikipedia.org/wiki/Burnside’s_lemma
For your problem, the group is the two-element group of flipping and the elements fixed by the non-identity element of the group are the “palindromic” patterns (same backwards and forwards) and (trivially) the identity element fixes everything.
To figure out who many total patterns there are, you have (let (x C y) mean (x choose y))
(42 C10)*(32 C 9)*(23 C 7)*(16 C 12)*(4 C 4).
Now you have to figure out how many are palindromic. But you are lucky – since there are an even number of beads, in order for a pattern to be palindromic, it must have all it’s color have an even number. So none of the patterns here can be palindromic.
Thus, by Burnside’s Lemma (which is trivial now, since it is just noticing that each necklace forms 2 patterns distinct patterns and so you are overcounting by a factor of 2), the total number of necklaces is:
(1/2)*(42 C10)*(32 C 9)*(23 C 7)*(16 C 12)*(4 C 4)
Anyway, it’s a huge number, so I’ll leave it like that (if you want to calculate it, be my guest).
Slide My “Mother Earth” Necklace
|
|
1x CZ STERLING SILVER FLOWER NECKLACE SLIDE PENDANT CONNECTOR #1231 $0.01 |
|
|
Vintage slide style necklace, white, pink, green, blue $0.50 |
|
|
925 Sterling authentic natural 11 mm South Sea Pearl chain necklace slide NR $0.95 |
|
|
CUTE LITTLE GOLD TONE AVON *TURTLE* SLIDE NECKLACE 98c NR $0.98 |
|
|
GOLDTONE HERRINGBONE CHAIN NECKLACE BLACK ONYX GLASS STONE SLIDE PENDANT $0.99 |
|
|
Vintage~ AVON ~ Abalone Shell Slide Pendant NECKLACE ~ Silver tone ~ Pretty! $0.99 |
|
|
30pcs 10mm Heart Slide Charms Fit Pet Collar DIY Necklace & Bracelet $0.99 |
|
|
50pcs 8mm Mix style Bird Slide Charms Fit Pet Collar DIY Necklace & Bracelet $0.99 |
|
|
50pcs 8mm Enamel Bird Slide Charms Fit Pet Collar DIY Necklace & Bracelet $0.99 |
|
|
50pcs 8mm Airplane Slide Charms Fit Pet Collar DIY Necklace & Bracelet $0.99 |
|
|
15pcs Mix Style DIY Necklace Chain Fit Pendant & slide charms $0.99 |
|
|
1970s RETRO STERLING COLLAR NECKLACE W/ EGYPTIAN ENAMEL SLIDE PENDANT 12 GRAMS $0.99 |
|
|
VTG 1980′s Goldtone Necklace Slide Penadant with Mobe Pearl $0.99 |
|
|
130pcs 10mm A-Z DIY Chrome slide letter Fit Pet Collar Necklace & Wristband $0.99 |
|
|
50pcs 8mm Enamel Bird Slide Charms Fit Pet Collar DIY Necklace & Bracelet $0.99 |
|
|
50pcs 8mm Slide Charms Fit Pet Collar DIY Necklace & Bracelet $0.99 |
|
|
VINTAGE Necklace Fashion Jewelry BLUE AQUA DONUT Foil Art Glass Bead SLIDE ROPE $0.99 |
|
|
VTG EMERALD GREEN MALACHITE STONE MEXICO STERLING SILVER NECKLACE PENDANT SLIDE $0.99 |
|
|
Lot of 18 Black Silk Cord Necklaces with Metal Slide-32”-Fully Adjustable Length $0.99 |
|
|
Orig 35mm Film Slide Hilary Swank Simon Baker Necklace $1.50 |
|
|
Orig 35mm Film Slide Hilary Swank Simon Baker Necklace $1.50 |
|
|
Estate Costume Jewelry Necklace Pendant Slide Rhinestone Reversible 6062 $1.99 |
|
|
Silvertone Engraved Initial “W” Deco Slide Pendant Necklace $1.99 |
|
|
10mm Round Freshwater Cultured Pearl ANGEL Necklace Slide / Charm / Pendent $1.99 |
|
|
Vintage Avon Jewelry Goldtone Letter M Necklace Chain Monogram Slide Pendant $2.00 |
|
|
John Hardy Sterling Silver Necklace Slide Enhancer $2.00 |
|
|
20 WATER SLIDE NAIL ART TRANSFER DECALS BETTY WITH A NECKLACE 5/8 INCH $2.10 |
|
|
VINTAGE GOLD TONE RED SCALLOP LUCITE SLIDE NECKLACE & EARRINGS SIGNED TRIFARI $2.25 |
|
|
Claire’s Deco Revival Silvertone Slide Pendant Necklace $2.25 |
|
|
PURPLE STAR SLIDE ON CHARM FOR BRACELET / NECKLACE $2.37 |