IIT JEE , AIEEE Forum - Mathematics , Algebra - Contest [swordfish #3]: Find ways to distribute identical balls in identical boxes

ayush007

the condition clearly says that no box remain empty.

hence , here a simple logic may do the job.

first of all, if i get the freedom of leaving the boxes empty, then the no. of ways would be =30*29*28. till here its going well.

now the thinking part,
(as the boxes are identicals hence no mater which is filled completly).

the ways in which the boxes can be left empty are ------

1. when one box is completely filled with all the balls.
2. when one box has 29 balls and any one of them is filled with 1 ball.
3.now, we should fill only two boxes with all the balls. right!!!
so,
ways to fill two boxes =30*29(which have 2nd case inclusive)

now, we have the calculation part left only,
( 30*29*28 )-( 30*29 )-1=30*29*27-1=23490-1=23489
hence the ans. is 23489.

its time that the author reveals the answer,and if anone finds any flaw in my mrthod , please make me aware(though i've tried to be perferctly correct ). it would be a great help .

thanks a lot .

all of you ---keep smiling.......

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