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

is coefficient  of $x^{15}$ in $x^6{\left(1+x+x^2+\mathrm{.....}\infty \right)}^6$  if no bag remains empty

is coefficient of $x^{15}in{\left(1-x\right)}^{-6}$

is same as number of the integral solutions of $a+b+c+d+e+f=15$

is same as number of non-negative  integral solutions of  $\sum _{i=1}^6{x}_i=15$