Number of ways in which 15 identical coins can be put into 6 different bags

Options

A. is coefficient of `x^(15)` in `x^(6)(1+x+x^(2)+ . . .oo)^(6),` if no bag remains empty
B. is coefficient of `x^(15)` in `(1-x)^(-6)`
C. is same as number of the integral solutions of
`a+b+c+d+e+f=15`
D. is same as number of non-negative integral solutions of `underset(i=1)overset(6)(sum)x_(i)=15`

Answer

A, B, D

Explanation

Let bags be `x_(1),x_(2),x_(3),x_(4),x_(5) and x_(6)`, then `x_(1)+x_(2)+x_(3)+x_(4)+x_(5)+x_(6)=15`.
`therefore` For no bags remains empty, number of ways
=Coefficient of `x^(15)` in `(x^(1)+x^(2)+x^(3)+ . . Oo)^(6)`
=Coefficient of `x^(15)` in `x^(6)(1+x+x^(2)+ . . .oo)^(6)`
=Coefficient of `x^(9)` in `(1-x)^(-6)`
In option (c), it is not mentioned that solution is positive integral.