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2n-objects-of-each-of-three-kinds-are-given-to-two-persons-so-that-each-person-gets-3n-objects-Prove-that-this-can-be-done-in-3n-2-3n-1-ways-

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Question Number 170794 by mr W last updated on 30/May/22
2n objects of each of three kinds are given to two persons, so that each person gets 3n objects. Prove that this can be done in 3n^2 + 3n + 1 ways.
2nobjectsofeachofthreekindsaregiventotwopersons,sothateachpersongets3nobjects.Provethatthiscanbedonein3n2+3n+1ways.
Answered by aleks041103 last updated on 05/Jun/22
let the first person have a,b and c objects from the 1st, 2nd and 3rd kind ⇒a+b+c=3n 0≤a,b,c≤2n b+c=3n−a 1) for a=0,...,n (b,c)∈{(n−a,2n),...,(2n,n−a)} ⇒there are 2n−(n−a)+1=n+a+1 ways for given a ⇒ for all a=0,...,n we have in total: Σ_(a=0) ^n (n+1+a)=(n+1)^2 +((n(n+1))/2) 2) for a=n+1,...,2n (b,c)∈{(0,3n−a),...,(3n−a,0)} ⇒ there are 3n+1−a ways for given a ⇒for all a=n+1,...,2n, in tot.: Σ_(a=n+1) ^(2n) (3n+1−a)=Σ_(k=1) ^n (3n+1−(k+n))= =Σ_(k=1) ^n (2n+1−k)=(2n+1)(n)−((n(n+1))/2) 3) total: ways=(n+1)^2 +((n(n+1))/2)+(2n+1)n−((n(n+1))/2)= =(n+1)^2 +2n^2 +n= =n^2 +2n+1+2n^2 +n= =3n^2 +3n+1
letthefirstpersonhavea,bandcobjectsfromthe1st,2ndand3rdkind⇒a+b+c=3n0⩽a,b,c⩽2nb+c=3n−a1)fora=0,…,n(b,c)∈{(n−a,2n),…,(2n,n−a)}⇒thereare2n−(n−a)+1=n+a+1waysforgivena⇒foralla=0,…,nwehaveintotal:∑na=0(n+1+a)=(n+1)2+n(n+1)22)fora=n+1,…,2n(b,c)∈{(0,3n−a),…,(3n−a,0)}⇒thereare3n+1−awaysforgivena⇒foralla=n+1,…,2n,intot.:∑2na=n+1(3n+1−a)=∑nk=1(3n+1−(k+n))==∑nk=1(2n+1−k)=(2n+1)(n)−n(n+1)23)total:ways=(n+1)2+n(n+1)2+(2n+1)n−n(n+1)2==(n+1)2+2n2+n==n2+2n+1+2n2+n==3n2+3n+1
Commented by mr W last updated on 05/Jun/22
thanks sir!
thankssir!
Answered by mr W last updated on 06/Jun/22
(1+x+x^2 +x^3 +...+x^(2n) )^3 =(((1−x^(2n+1) )^3 )/((1−x)^3 )) =(1−3x^(2n+1) +3x^(4n+2) −x^(6n+3) )Σ_(k=0) ^∞ C_2 ^(k+2) x^k coefficient of x^(3n) term: C_2 ^(3n+2) −3C_2 ^(n+1) =(((3n+2)(3n+1)−3(n+1)(n))/2) =((9n^2 +9n+2−3n^2 −3n)/2) =3n^2 +3n+1
(1+x+x2+x3+…+x2n)3=(1−x2n+1)3(1−x)3=(1−3x2n+1+3x4n+2−x6n+3)∑∞k=0C2k+2xkcoefficientofx3nterm:C23n+2−3C2n+1=(3n+2)(3n+1)−3(n+1)(n)2=9n2+9n+2−3n2−3n2=3n2+3n+1
Commented by aleks041103 last updated on 05/Jun/22
This is very clever! Thank you!
Thisisveryclever!Thankyou!

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