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4 votes
4 votes
In how many different ways 10 identical ball can be distributed among 3 children, if each receives atleast 2 balls and no more than 4 balls?

2 Answers

Best answer
7 votes
7 votes

Whenever we have upper and lower bounds and the distribution of identical objects is done in distinct boxes/entities then the concept of generating function and multinomial theorem comes into picture..Hence we can write as :

No of solutions of the equation :  x1 + x2 + x3 = 10    where  2 <= x1,x2,x3 <= 4

So it is nothing but the coefficient of x10  in (x2  + x3  +  x4)3   =  x6 (1 + x + x2)3

                         ==>   coefficient of x4  in  (1 + x + x2)3        =  (1 - x3)3 (1 - x)-3

Now we know :

Coefficient of xr  in  (1 - x)-n    =   n-1+rCr

So we will get relevant coefficients using "1" and "-3x3 " term of   (1 - x3)3  only as other exceed x4 ..

                       ==>   coefficient of x4  in  (1 + x + x2)3      =  1 * (3-1+4C4 )   -  3 * 3-1+1C1

                                                                                      =  15   -  9

                                                                                      =  6

Hence required number of ways of distribution    =    6

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3 votes
3 votes
I think simple approach would be easier since numbers are small,

10 balls, 3 kids

1. For atleast getting 2 balls each distribute 2 balls each to 3 kids

2+2+2=6

Now remaining balls 10-6=4 balls

distribute 4 balls among 3 kids there are 4+3-1 c 4  =15 ways

But 2nd condition says not more than 4 balls for each kid

This will happen if we distribute 4 or 3 balls to any kid from 4 available balls since each already has 2 balls each

i.e 2+4=6 avoid this case

or 2+3=5 avoid this case.

Therefore 4 balls to any kid will have 3 ways and 3 balls to any kid will have 6 ways i.e 9 ways to avoid

Hence 15-9=6 ways

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