Difference between revisions of "User:Ryanbear"
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| + | 127404754587834447066121402544149752403367309839754465976733876850535259518475909594695860306985322205112937748580959757733111433157235701022282968119850836745546278307667704653372658552406000753491774551479865697792135477161195521056948854517854321252265742150501216164186030139871809346990872684920506980997357017966298200142758044809690492762992755660707017875279058967437007588929796277764060005089298064214879589772413549606947968900674428405521400284241554092224188812033614398140341416473369322228003712839254347683876360616572060502059408555377390203293482422877390956933461744147435089970769638752323062428466355987800018937157222463250323840029581098946400823076380412044514339147538180167529051650422579358398763747733709530588882192847356109601997103404801625719264806184704255207773908632121628121218556422577808407442065598527465689061110360187188596406144329463393282277302397966695581626834527593057345685374346326761582158844627492731128776756877171629235767714578716392603611050291611235105023611651718910803700546140521204265462337603808882061589065658546143476080287873475025789654838047330757693980821316863367969599435264122353134402662193709738448765684330622112034563823693091508527188449444500442756555017615788366729759464961260910397767745133388131924611118524177500315369815288246600110733944359683619713264458796518204141705157408836206828853802276808129896518197206059000967489007947494199187456685532519042382615340141998121997925402526097853864089771736576331094606267537797084686987144893315555255778802812164754555272187462332469488499359068615277933740654314904814067621603997908775954281427807648276302695080017958941502363864647300207518768563640133589122519343502638958821545894862462914776956651328199557013757056920367625244333556095179493403092211525722809465689794903682679719038165781179884534083331850224305649971759849018003101612781711795572282485772318031665207271428539033496032608058902377993956463498837626382048313745553856555641257865927315980059368278785925298640945249848925204586323353123125638120957640808875972082926783468636960979905910145024 | ||
| + | ==stars and bars on differences== | ||
| + | To find the number of ways to choose <math>n</math> numbers between <math>a</math> and <math>b</math> if they are indistinguishable but can have duplicates let <math>a_1</math> be the difference between the smallest number and <math>a</math>, <math>a_2</math> be the difference between the 2rd and 1st smallest numbers, and similar logic to get <math>a_{n+1}</math> is the difference between <math>b</math> and the largest number. <math>a_1+a_2+...+a_{n+1}=b-a</math>. Then use stars and bars to get <math>{n+b-a \choose n}</math> | ||
Latest revision as of 12:39, 22 October 2023
127404754587834447066121402544149752403367309839754465976733876850535259518475909594695860306985322205112937748580959757733111433157235701022282968119850836745546278307667704653372658552406000753491774551479865697792135477161195521056948854517854321252265742150501216164186030139871809346990872684920506980997357017966298200142758044809690492762992755660707017875279058967437007588929796277764060005089298064214879589772413549606947968900674428405521400284241554092224188812033614398140341416473369322228003712839254347683876360616572060502059408555377390203293482422877390956933461744147435089970769638752323062428466355987800018937157222463250323840029581098946400823076380412044514339147538180167529051650422579358398763747733709530588882192847356109601997103404801625719264806184704255207773908632121628121218556422577808407442065598527465689061110360187188596406144329463393282277302397966695581626834527593057345685374346326761582158844627492731128776756877171629235767714578716392603611050291611235105023611651718910803700546140521204265462337603808882061589065658546143476080287873475025789654838047330757693980821316863367969599435264122353134402662193709738448765684330622112034563823693091508527188449444500442756555017615788366729759464961260910397767745133388131924611118524177500315369815288246600110733944359683619713264458796518204141705157408836206828853802276808129896518197206059000967489007947494199187456685532519042382615340141998121997925402526097853864089771736576331094606267537797084686987144893315555255778802812164754555272187462332469488499359068615277933740654314904814067621603997908775954281427807648276302695080017958941502363864647300207518768563640133589122519343502638958821545894862462914776956651328199557013757056920367625244333556095179493403092211525722809465689794903682679719038165781179884534083331850224305649971759849018003101612781711795572282485772318031665207271428539033496032608058902377993956463498837626382048313745553856555641257865927315980059368278785925298640945249848925204586323353123125638120957640808875972082926783468636960979905910145024
stars and bars on differences
To find the number of ways to choose 
numbers between 
and 
if they are indistinguishable but can have duplicates let 
be the difference between the smallest number and , 
be the difference between the 2rd and 1st smallest numbers, and similar logic to get 
is the difference between and the largest number. 
. Then use stars and bars to get 
