In how many different ways can 3 persons A, B, C having 6 one-rupee coin 7 one-rupee coin, 8 one-rupee coin, respectively, donate 10 one-rupee coin collectively?

Explanation

Let A, B and C donate `x_(1),x_(2),x_(3)` number of one-rupee coins.
`therefore x_(1)+x_(2)+x_(3)=10`, where `0 le x_(1) le 6, 0 le x_(2) le 7, 0 le x_(3) le 8`
` therefore` Required number of ways =coefficient to `p^(10) " in " (1+p+p^(2)+..+p^(6))xx(1+p+p^(2)+..+p^(7))xx(1+p+p^(2)+..+p^(8))`
(In each bracket series is not extended to infinite terms as upper limit of each variable is less than 10)
=coefficient to `p^(10) " in " ((1-p^(7))/(1-p))((1-p^(8))/(1-p))((1-p^(9))/(1-p))`
=coefficient of `p^(10) " in " (1-p^(7))(1-p^(8))(1-p^(9))(1-p)^(-3)`
=coefficient of `p^(10) " in " (1-p^(7)-p^(8)-p^(9))xx(1+ ""^(3)C_(1)p+ ""^(4)C_(2)p^(2)+ ""^(5)C_(3)p^(3)+..+ ""^(12)C_(10)p^(10)+..)`
`=""^(12)C_(10)- ""^(5)C_(3)- ""^(4)C_(2)- ""^(3)C_(1)`
=66-10-6-3
=47