dice <- data.frame(
  "6" = choose(6,6)*choose(6,1),
  "51" = choose(6,5)*choose(6,1)*choose(1,1)*choose(5,1),
  "42" = choose(6,4)*choose(6,1)*choose(2,2)*choose(5,1),
  "411" = choose(6,4)*choose(6,1)*choose(2,1)*choose(1,1)*choose(5,2),
  "33" = choose(6,3)*choose(3,3)*choose(6,2),
  "321" = choose(6,3)*choose(6,1)*choose(3,2)*choose(5,1)*choose(1,1)*choose(4,1),
  "3111" = choose(6,3)*choose(6,1)*choose(3,2)*choose(5,1)*choose(1,1)*choose(4,1),
  "222" = choose(6,2)*choose(4,2)*choose(2,2)*choose(6,3),
  "2211" = choose(6,2)*choose(4,2)*choose(6,2)*choose(2,1)*choose(1,1)*choose(4,2),
  "21111" = choose(6,2)*choose(6,1)*choose(4,1)*choose(3,1)*choose(2,1)*choose(1,1)*choose(5,4),
  "111111" = choose(6,1)*choose(5,1)*choose(4,1)*choose(3,1)*choose(2,1)*choose(1,1)*choose(6,6)
)
sum(dice) - 6^6
dice/6^6