lets assume 6 boxes A,B,C,D,E,F
first two boxes can have k balls where 0≤k≤4
A+B=k
so Total ways of distributing n similar objects between r boxes is $^{_{r-1}^{n+r-1}\textrm{C}}$
so $^{_{2-1}^{k+2-1}\textrm{C}}$ = $^{_{1}^{k+1}\textrm{C}}$ =(k+1)
now distribute remaining 10-k objects in the remaining 4 boxes
C+D+E+F=10-k
$^{_{3}^{13-k}\textrm{C}}$
now multiply both the terms and apply summation for k=0 to 4
$\sum_{k=0}^{4} (k+1)_{3}^{13-k}\textrm{C}$