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Number-of-distributions-of-n-different-articles-to-r-different-boxes-so-as-1-empty-box-allowed-2-empty-box-not-allowed-with-proof-kindly-help-me-




Question Number 195538 by SLVR last updated on 04/Aug/23
Number of distributions of  n different articles to r different  boxes  so as 1)empty box allowed  2)empty box not allowed  with proof...kindly help me
Numberofdistributionsofndifferentarticlestordifferentboxessoas1)emptyboxallowed2)emptyboxnotallowedwithproofkindlyhelpme
Commented by SLVR last updated on 04/Aug/23
sir...kindly give proof
sirkindlygiveproof
Commented by SLVR last updated on 05/Aug/23
sir Mr.W..or...any one can  help me...i need bit urgently
sirMr.W..oranyonecanhelpmeineedbiturgently
Commented by mr W last updated on 05/Aug/23
using generating function method  we can see that the result is the  coefficient of term x^n  in expansion  1) n!(1+(x/1)+(x^2 /(2!))+(x^3 /(3!))+...)^r , i.e.  n!e^(rx)   2) n!((x/1)+(x^2 /(2!))+(x^3 /(3!))+...)^r , i.e.  n!(e^x −1)^r
usinggeneratingfunctionmethodwecanseethattheresultisthecoefficientoftermxninexpansion1)n!(1+x1+x22!+x33!+)r,i.e.n!erx2)n!(x1+x22!+x33!+)r,i.e.n!(ex1)r
Commented by SLVR last updated on 05/Aug/23
Thanks Prof.W...so kind of you
ThanksProf.Wsokindofyou
Commented by mr W last updated on 05/Aug/23
a little explanation:  say box 1 has k_1  objects, box 2 has k_2   objects, etc. with k_1 +k_2 +...k_r =n and  k_1 ,k_2 ,...,k_r  are fixed values.  to distribute n objects in r boxes in  this way there are ((n!)/((k_1 )!×(k_2 )!×...(k_r )!))  possibilities.  since k_1 ,k_2 ,...,k_r  can be variable values,  we can construct a generating  function like  n!((x/(1!))+(x^2 /(2!))+...+...)^r  if each box must  get at least an object, or  n!(1+(x/(1!))+(x^2 /(2!))+...+...)^r  if each box may  also be empty.    for 1) the formula is r^n .  for 2) the formula is      r^n −C_1 ^r (r−1)^n +C_2 ^r (r−2)^n −...+(−1)^(n−1) C_(r−1) ^r
alittleexplanation:saybox1hask1objects,box2hask2objects,etc.withk1+k2+kr=nandk1,k2,,krarefixedvalues.todistributenobjectsinrboxesinthiswaytherearen!(k1)!×(k2)!×(kr)!possibilities.sincek1,k2,,krcanbevariablevalues,wecanconstructageneratingfunctionliken!(x1!+x22!++)rifeachboxmustgetatleastanobject,orn!(1+x1!+x22!++)rifeachboxmayalsobeempty.for1)theformulaisrn.for2)theformulaisrnC1r(r1)n+C2r(r2)n+(1)n1Cr1r
Commented by mr W last updated on 05/Aug/23
example:  we have 10 foreign students who  should be distributed to 3 schools.  in how many ways can this be done?  1) 3^(10) =59049 ways  2) 3^(10) −3×2^(10) +3×1^(10) =55980 ways
example:wehave10foreignstudentswhoshouldbedistributedto3schools.inhowmanywayscanthisbedone?1)310=59049ways2)3103×210+3×110=55980ways
Commented by mr W last updated on 05/Aug/23
Answered by MM42 last updated on 05/Aug/23
if  n≥r  let  x_i  = number articles in the i^(th)  box  1)⇒ x_1 +x_2 +...+x_r =n  ;  x_i  ≥0  ans= (((n+r−1)),((         n)) )=(((n+r−1)!)/(n!(r−1)!))  2)⇒ x_1 +x_2 +...+x_r =n  ;  x_i  ≥1  ⇒ y_1 +y_2 +...+y_r =n−r  ;  y_i  ≥0  ans= (((n−1)),((n−r)) )=(((n−1)!)/((n−r)!(r−1)!))  if  n<r  then  no all box can be full.
ifnrletxi=numberarticlesintheithbox1)x1+x2++xr=n;xi0ans=(n+r1n)=(n+r1)!n!(r1)!2)x1+x2++xr=n;xi1y1+y2++yr=nr;yi0ans=(n1nr)=(n1)!(nr)!(r1)!ifn<rthennoallboxcanbefull.
Commented by mr W last updated on 05/Aug/23
what you solved is for the case that  the objects are identical. but the  question said that both the boxes  and the objects are different.
whatyousolvedisforthecasethattheobjectsareidentical.butthequestionsaidthatboththeboxesandtheobjectsaredifferent.
Commented by MM42 last updated on 08/Aug/23
yes.you are right.i did not notice  the diffrence between the subjects.  thanks for you
yes.youareright.ididnotnoticethediffrencebetweenthesubjects.thanksforyou

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