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In how many ways 3 friends ram ,rajat and rupesh having 6 one rupee coins, 7 one rupee coins ,8 one rupee coins respectively donate 10 rupees collectively.

a)40

b)45

c)47

d)50
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The given problem can be formulated as :

x1 + x2 + x3 = 10 subjected to constraints :

0 <= x1 <= 6 ,

0 <= x2 <= 7

0 <= x3 <= 8

So solution to the given problem is nothing but finding coefficient of x10  in (1 + x ..+ x6) . (1 + x .. + x7) . ( 1 + x ....+ x8)

                                                         ==>   coefficient of x10  in  (1 - x7) . (1 - x8) . (1 - x9 ) .( 1 - x ) -3

Now out of the terms of (1 - x7) . (1 - x8) . (1 - x9 ) , we extract only those terms which contribute to x10 ..

So

(1 - x7 ) . (1 - x8 - x9 + x17 )    =  1  -  x - x9  -  x7  [ Only those terms which contribute to x10 are mentioned ]..

Now for each of the 4 terms we find the corresponding coefficient in ( 1 - x ) -3  and then finally sum up to find the answer.

So due to the term 1 , we have :   3-1+10C10   =   66

      due to the term 2 , we have :  3-1+2C2  =  6

        due to the term 2 , we have :  3-1+1C1  =  3

        due to the term 2 , we have :  3-1+3C3  = 10

Hence , no of ways  =   66  -  6 - 3 - 10 

                              =   47

Hence 47 should be the correct answer.

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