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

johri_anshuman

As each ball can be distributed each in one cylinder thus the total ways = 3*3*3.....30 times=3³⁰.

stareawe

you said 30 identical balls are given then for writing iuse "0" for a ball . Three identical boxes are given .I use the following method to find the answer:

I take two separater marks(/)and arrange the 0's inside them and outside them like
00/00.......0/00..0 ie., the 0's(no. of balls) before the separater marks go into the go into the first box and likewise. since no box should be empty, i club the first'/' with a 0 ie.,(0/) and secon with two 0's hence i have 27 identical balls and two distinct elements hence the no. of ways

= 29!/27!
= 812

aspiredtolive

m pretty sure bout my answer....
plz check it on page 4....
n gimme a reply soon plz

fatafati

chander_singla06

1+1+1 in three boxes

now remaining 27 balls have to be placed in 3 itdetical boxes

27

3

----------------------------

3 !

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.......

Arpit_Rocks

ayush007

hey, please give the answer.
what happened to the admin, he has not even given a clue of his presence.......
waiting.................................................................................

keep smiling

sg_90

use the multinomial theorem taking x1,x2&x3

and find the no.of sl. to the equatin x1+x2+x3=30

where each variable is ranging from 1to 27

the answer is 476

aysh

it's simple...
if the no. of balls in each of the boxes is denoted by x, y, z, then
x+ y+ z=30 (where, x,y,z>0 ; x,y,z are integers)
this gives the number of ways to be
171.
but the boxes are identical.
so reqd no. of ways=171/3=57

aysh

sorry! went wrong,
the answer is 33.

ayu

i agree with ayush007. great dude...........

master_purav

ririlok

the ans is28 ways

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