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Question Number 21802 by Tinkutara last updated on 04/Oct/17
Five balls are to be placed in three  boxes. Each can hold all the five balls.  In how many different ways can we  place the balls so that no box remains  empty, if balls are different but boxes  are identical?
$$\mathrm{Five}\:\mathrm{balls}\:\mathrm{are}\:\mathrm{to}\:\mathrm{be}\:\mathrm{placed}\:\mathrm{in}\:\mathrm{three} \\ $$$$\mathrm{boxes}.\:\mathrm{Each}\:\mathrm{can}\:\mathrm{hold}\:\mathrm{all}\:\mathrm{the}\:\mathrm{five}\:\mathrm{balls}. \\ $$$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{different}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{we} \\ $$$$\mathrm{place}\:\mathrm{the}\:\mathrm{balls}\:\mathrm{so}\:\mathrm{that}\:\mathrm{no}\:\mathrm{box}\:\mathrm{remains} \\ $$$$\mathrm{empty},\:\mathrm{if}\:\mathrm{balls}\:\mathrm{are}\:\mathrm{different}\:\mathrm{but}\:\mathrm{boxes} \\ $$$$\mathrm{are}\:\mathrm{identical}? \\ $$
Commented by mrW1 last updated on 06/Oct/17
25 ways
$$\mathrm{25}\:\mathrm{ways} \\ $$
Answered by ajfour last updated on 06/Oct/17
coeff. of x^5  in ((5!)/(3!))[x+(x^2 /(2!))+(x^3 /(3!))]^3   =coeff. of x^2  in 20(1+(x/2)+(x^2 /6))^3      =20((3/6)+(3/4)) =20×(5/4)=25 .
$${coeff}.\:{of}\:{x}^{\mathrm{5}} \:{in}\:\frac{\mathrm{5}!}{\mathrm{3}!}\left[{x}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}!}+\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}\right]^{\mathrm{3}} \\ $$$$={coeff}.\:{of}\:{x}^{\mathrm{2}} \:{in}\:\mathrm{20}\left(\mathrm{1}+\frac{{x}}{\mathrm{2}}+\frac{{x}^{\mathrm{2}} }{\mathrm{6}}\right)^{\mathrm{3}} \\ $$$$\:\:\:=\mathrm{20}\left(\frac{\mathrm{3}}{\mathrm{6}}+\frac{\mathrm{3}}{\mathrm{4}}\right)\:=\mathrm{20}×\frac{\mathrm{5}}{\mathrm{4}}=\mathrm{25}\:. \\ $$
Commented by Tinkutara last updated on 06/Oct/17
But I don′t understand why to find  such coefficients in these expansions?  I know this formula but don′t  understand the reason.
$$\mathrm{But}\:\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{understand}\:\mathrm{why}\:\mathrm{to}\:\mathrm{find} \\ $$$$\mathrm{such}\:\mathrm{coefficients}\:\mathrm{in}\:\mathrm{these}\:\mathrm{expansions}? \\ $$$$\mathrm{I}\:\mathrm{know}\:\mathrm{this}\:\mathrm{formula}\:\mathrm{but}\:\mathrm{don}'\mathrm{t} \\ $$$$\mathrm{understand}\:\mathrm{the}\:\mathrm{reason}. \\ $$
Commented by ajfour last updated on 09/Oct/17
if x_1 +x_2 +x_3 =9  placing 9 balls in three boxes such  that they can remain empty, then  it is equivalent to  choosing x^9  from  the expansion of  (1+x+x^2 +...)^3   =(1+x+x^2 +...)(1+x+x^2 +...)(1+x+x^2 +..)  because powers of x from each  bracket shall add and similarly  no. of balls in each box shall add.  the sum has to be 9 , so we choose  no. of ways of obtaining x^9  =  coefficient of x^9  .  =coeff. of x^9  in the expansion  of (1+x+x^2 +...)^3   or in (1−x)^(−3)  =^(9+3−1) C_(3−1) =^(11) C_2  .
$${if}\:{x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} =\mathrm{9} \\ $$$${placing}\:\mathrm{9}\:{balls}\:{in}\:{three}\:{boxes}\:{such} \\ $$$${that}\:{they}\:{can}\:{remain}\:{empty},\:{then} \\ $$$${it}\:{is}\:{equivalent}\:{to}\:\:{choosing}\:{x}^{\mathrm{9}} \:{from} \\ $$$${the}\:{expansion}\:{of} \\ $$$$\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +…\right)^{\mathrm{3}} \\ $$$$=\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +…\right)\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +…\right)\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +..\right) \\ $$$${because}\:{powers}\:{of}\:{x}\:{from}\:{each} \\ $$$${bracket}\:{shall}\:{add}\:{and}\:{similarly} \\ $$$${no}.\:{of}\:{balls}\:{in}\:{each}\:{box}\:{shall}\:{add}. \\ $$$${the}\:{sum}\:{has}\:{to}\:{be}\:\mathrm{9}\:,\:{so}\:{we}\:{choose} \\ $$$${no}.\:{of}\:{ways}\:{of}\:{obtaining}\:{x}^{\mathrm{9}} \:= \\ $$$${coefficient}\:{of}\:{x}^{\mathrm{9}} \:. \\ $$$$={coeff}.\:{of}\:{x}^{\mathrm{9}} \:{in}\:{the}\:{expansion} \\ $$$${of}\:\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +…\right)^{\mathrm{3}} \\ $$$${or}\:{in}\:\left(\mathrm{1}−{x}\right)^{−\mathrm{3}} \:=\:^{\mathrm{9}+\mathrm{3}−\mathrm{1}} {C}_{\mathrm{3}−\mathrm{1}} =^{\mathrm{11}} {C}_{\mathrm{2}} \:. \\ $$
Commented by mrW1 last updated on 06/Oct/17
nice explained!    as result one can use the notation of  stirling number {_3 ^5 } which means  the number of ways to put 5 different  objects into 3 identical boxes.  {_3 ^5 }=25
$$\mathrm{nice}\:\mathrm{explained}! \\ $$$$ \\ $$$$\mathrm{as}\:\mathrm{result}\:\mathrm{one}\:\mathrm{can}\:\mathrm{use}\:\mathrm{the}\:\mathrm{notation}\:\mathrm{of} \\ $$$$\mathrm{stirling}\:\mathrm{number}\:\left\{_{\mathrm{3}} ^{\mathrm{5}} \right\}\:\mathrm{which}\:\mathrm{means} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{to}\:\mathrm{put}\:\mathrm{5}\:\mathrm{different} \\ $$$$\mathrm{objects}\:\mathrm{into}\:\mathrm{3}\:\mathrm{identical}\:\mathrm{boxes}. \\ $$$$\left\{_{\mathrm{3}} ^{\mathrm{5}} \right\}=\mathrm{25} \\ $$
Commented by Tinkutara last updated on 06/Oct/17
Thank you very much Sir!
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$
Commented by mrW1 last updated on 09/Oct/17
to mr. ajfour:  you said:  =coeff. of x^9  in the expansion  of (1+x+x^2 +...)^3   or in (1−x)^(−3)  =^(9−3+1) C_3 =^7 C_3  .    should the last line not be:  or in (1−x)^(−3)  =^(9+3−1) C_9 =^(11) C_9  .  ?
$$\mathrm{to}\:\mathrm{mr}.\:\mathrm{ajfour}: \\ $$$$\mathrm{you}\:\mathrm{said}: \\ $$$$={coeff}.\:{of}\:{x}^{\mathrm{9}} \:{in}\:{the}\:{expansion} \\ $$$${of}\:\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +…\right)^{\mathrm{3}} \\ $$$${or}\:{in}\:\left(\mathrm{1}−{x}\right)^{−\mathrm{3}} \:=\:^{\mathrm{9}−\mathrm{3}+\mathrm{1}} {C}_{\mathrm{3}} =^{\mathrm{7}} {C}_{\mathrm{3}} \:. \\ $$$$ \\ $$$$\mathrm{should}\:\mathrm{the}\:\mathrm{last}\:\mathrm{line}\:\mathrm{not}\:\mathrm{be}: \\ $$$${or}\:{in}\:\left(\mathrm{1}−{x}\right)^{−\mathrm{3}} \:=\:^{\mathrm{9}+\mathrm{3}−\mathrm{1}} {C}_{\mathrm{9}} =^{\mathrm{11}} {C}_{\mathrm{9}} \:. \\ $$$$? \\ $$
Commented by ajfour last updated on 09/Oct/17
yes it is in fact as you say..  i dint remember correctly, n   thought i did.
$${yes}\:{it}\:{is}\:{in}\:{fact}\:{as}\:{you}\:{say}.. \\ $$$${i}\:{dint}\:{remember}\:{correctly},\:{n}\: \\ $$$${thought}\:{i}\:{did}. \\ $$

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