July 7 - Marbles and Salt Shakers

The problem is equivalent to Will dropping  $m=12$  marbles into  $s=3$  salt shakers with the condition that each salt shaker must contain at least one marble. By the marbles and salt shakers theorem, the number of ways for Will to paint his sticks is  $\binom{m-1}{s-1}=\binom{11}{2}=55$ .

Same problem as before, except we have removed the condition that each salt shaker needs a marble. By the other marbles and salt shakers theorem, the number of ways for Roman to paint his sticks is  $\binom{m+s-1}{s}=\binom{14}{3}=364$ .