 yea bhaagam aadhaaramga combinatorics ‌ loo burn ‌ seide lemma yokka konni aacharanaatmaka anuvartanaalu emti ?
 burn ‌ seide lemma ,  konnisarlu burn ‌ seide yokka lekkimpu siddhaantam ,  kauchi - frobenius lemma ,  arbit - counting siddhaantam ledha burn ‌ seide laeni lemma ani kudaa pilustharu ,  idi samooha siddhaantam falithamgaa ganita vastuvulanu lekkinchetappudu samaroopatanu parigananaloki teesukovadamlo tarachugaa vupayogapaduthundi .  deeni vividha perlu viliam burn ‌ seide ,  gorge polya ,  agustin luis kauchi mariyu ferdinand gorge frobenius ‌ lapai aadhaarapadi unnayi .  phalitham burn ‌ seide will kadhu ,  athanu dhaanini tana pustakam  ' aan dhi thierry af groops af finite aurdar ' loo utamkistuu ,  dhaanini frobenius  (1887) ki aapaadinchaadu .  burn ‌ seide yokka lemma  " kakshyalanu "  ganistundi ,  idi samaroopatanu parigananaloki tisukuni vibhinna vastuvulanu lekkinchadam vantidhe .  dheenini cheppe itara margalu samaanamaina sambandam  R  varku vibhinna vastuvulanu lekkinchadam ledha niyamaanuguna ruupamloe unna vastuvulanu lekkinchadam .

 kindhi vatilo , G  ooka sett  X pai panichaesae parimitha samuhamga undanivvandi . G loni prathi  g  choose , X loni muulakaala samitini  X  dwara suuchimchanivvandi ,  adi  g  dwara sthiramgaa umtumdi  (g  dwara maarpulenidigaa kudaa cheppabaduthundi ),  antey  , Xg = { x ∈ X | g.x = x }.  burn ‌ seide lemma kakshyala sanka choose krindhi suutraanni nirdhaaristundi , |X/G|:

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{\displaystyle |X/G|={\frac {1}{|G|}}\sum _{g\in G}|X^{g}|.}
 amduvalana kakshyala sanka  ( ooka sahaja sanka ledha  +∞) G  yokka moolakam dwara sthireekarinchabadina paayimtla sagatu sankhyaku samaanam  ( idi kudaa sahaja sanka ledha anantam ). G  anantam ayithe , |G|  dwara vibhajana bagaa nirvachinchabadakapovachchu ;  yea sandarbhamlo cardinal ankaganitamlo kindhi prakatana umtumdi :

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{\displaystyle |G||X/G|=\sum _{g\in G}|X^{g}|.}
 gananaku darakhaastula udaaharanalu 
 neckless ‌ lu 
3  podavu gala bitt vectors  8  saadhyamee ,  conei  3: 000, 001, 011  mariyu  111  podavu gala nalaugu vibhinna  2- rangu neckless ‌ lu Bara unnayi ,  endhukante  100  mariyu  010  bhramanam dwara  001 ki samaanam ;  adevidhamgaa  110  mariyu  101 011 ki samaanam .  phaarmulaa bhramanaala sankhyapai aadhaarapadi umtumdi ,  yea sandarbhamlo  3 ( suunya bhramanamtho sahaa )  mariyu prathi bhramanaaniki marani bitt vectors sanka .  motham  8  bitt wectar ‌ lu suunya bhramanam dwara maaravu mariyu remdu  (000  mariyu  111)  itara remdu bhramanaala dwara maaravu ,  ivi :
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{\displaystyle 4={\frac {1}{3}}(8+2+2)}.

 podavu  4  choose , 16  sadhyam bitt vectors unnayi ; 4  bhramanaalu ;  suunya bhramanam anni  16  bitt wectar ‌ lanu marchakunda vadilivestundi ; 1- bhramanam mariyu  3- bhramanam okkokkati remdu bitt wector ‌ lanu marchakunda vadilivestaayi  (0000  mariyu  1111); 2- bhramanam  4  bitt wectar ‌ lanu marchakunda vadilivestundi  (0000, 0101, 1010  mariyu  1111);  ivvadam :
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{\displaystyle 6={\frac {1}{4}}(16+2+4+2)}.  avi : 0000, 0001, 0011, 0101, 0111  mariyu  1111.

 ooka queb yokka rangulu 
 muudu rangulanu upayoginchi ooka queb yokka mukhaala yokka bhramanha vibhinna rangula sankhyanu yea phaarmulaa nundi yea krindhi vidhamgaa nirnayinchavachhu .

X  anede ooka nirdishta dhooraniloo queb ‌ ku vartinche  36  mukha rangu kalayikala samitigaa undanivvandi mariyu queb yokka bhramanha samuham  G  sahaja paddhatilo  X pai pania cheyanivvandi . X  yokka remdu muulakaalu okadhanikokati bhramanam aynappudu khachitamgaa oche kakshyaku chendinavi .  bhramanha vibhinna rangula sanka kakshyala sankhyaku samaanamgaa umtumdi mariyu  G  yokka  24  muulakaala choose sthiira sett ‌ l parimaanaalanu lekkinchadam dwara kanugonavacchu .


 rangu mukhaalato queb 
X  yokka motham  36  muulakaalanu marchakunda unchae ooka gurthimpu moolakam 
 aaru  90- degreela mukha bhramanaalu ,  veetilo prathi okkati  X  yokka  33  muulakaalu maaravu 
 muudu  180- degreela mukha bhramanaalu ,  veetilo prathi okkati  X  yokka  34  muulakaalanu maarchaledu 
 yenimidhi  120- degreela sheersha bhramanaalu ,  veetilo prathi okkati  X  yokka  32  muulakaalanu maarchaledu 
 aaru  180- degreela amchu bhramanaalu ,  veetilo prathi okkati  X  yokka  33  muulakaalu maaravu 
 yea automarpism ‌ l yokka vivaranaatmaka pariseelana ikda chudavachu .

 sagatu parishkaara parimaanam yea vidhamgaa umtumdi 

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57.
{\frac {1}{24}}\left(3^{6}+6\cdot 3^{3}+3\cdot 3^{4}+8\cdot 3^{2}+6\cdot 3 ^{3}\ Kandla )=57.
 anevalla muudu rangulaloo ooka queb mukhaala yokka  57  bhramanha vibhinna rangulu unnayi .  saadharanamga , n  rangulaloo ooka queb mukhaala yokka bhramanha vibhinna rangula sanka deeni dwara ivvabaduthundi 

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{\frac {1}{24}}\left(n^{6}+3n^{4}+12n^{3}+8n^{2}\right).
8  quiins puzzle 
 yenimidhi quiins puzzle ‌ loo  92  parishkaaralu unnayi ,  veetilo  12  praadhimika parishkaaralu boardu yokka bhramanha mariyu prathibimbam varku vibhinnamgaa untai .  suunya charyatho sahaa  8  bhramanaalu mariyu pratibimbaala kalayikalu unnayi .  suunya carya motham  92  parishkaaralanu marchakunda vadilivestundi . 92  parishkaaraalalo nalaugu sushtamgaa untai , 180°  bhramanamlo maarpu undadhu .  idi estunde :
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{\displaystyle 12={\frac {1}{8}}(92+4)}.
*  neckless lekkimpu samasya :  prathi pusa remdu rangulaloo okatiga umdae vividha podavula bhramanha vishisht neckless ‌ lanu lekkinchadam .  vyasam  3  mariyu  4  podavu gala neckless ‌ lanu lekkinchindi mariyu samadhanam varusaga  4  mariyu  6.
*  queb samasya yokka rangulu :  queb mukhaala bhramanha vibhinna rangulanu lekkinchandi .  vyasam  3  rangulanu upayoginchinappudu mariyu samadhanam  57  aynappudu ooka sandarbhaanni pariseelistundi .
* 8  quiins puzzle :  samasya yokka classic variation  8  chadaramgam kueen ‌ lanu  8x8  boardulo unchadaaniki vividha maargaalanu lekkinchadam ,  tadwara e raanee marokaripai daadi cheeyadu  ( samadhanam  92).  burn ‌ seide yokka lemma aplication ‌ thoo kathanam bhramanam mariyu prathibimbam varku kueen ‌ l vividha places ‌ ment ‌ l sankhyanu ganistundi ,  samaadhaanangaa  12 ki cherukuntundhi .