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$3$ boxes have some balls, for example box $A$ has $1$ balls, box $B$ has $2$ balls, box $C$ has $3$ balls

If only care the generating function of box $A$, can it use box $A$ and non box $A$ to count?

How to count? polynomial count?

Update: the number of balls in each box is determined by the number after throw a die, for box A, i throw a die, if get 1 then i put a ball in box A, for box B, if throw a die and get 2 then i put two balls in box B, etc..the goal is to concern the distribution in box A in a group of boxes

Remark: use z represent the number of appearance of balls in box A

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What exactly are you trying to count? –  Jeremy Jan 19 '13 at 7:58
    
count balls in box A –  Machine Gun Jan 19 '13 at 8:02
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i remember it's like something x+x^2+x^3 though i do not understand –  Machine Gun Jan 19 '13 at 8:03
    
What restrictions are there on the numbers of balls in each box? And no, you probably can't just consider boxes $B$ and $C$ as "one box". –  vonbrand Jan 21 '13 at 3:04
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