July 7 - Marbles and Salt Shakers

July 7 - Marbles and Salt Shakers

Roman Hall

Marbles and Salt Shakers

• The number of ways to drop  $m$  marbles into  $s$  salt shakers is given by  $\binom{m+s-1}{s}=\binom{m+s-1}{m-1}$  

• If we require that each salt shaker contain at least one marble, then the number of ways to drop the  $m$  marbles into the  $s$  salt shakers is given by  $\binom{m-1}{s-1}$ 

Open Ended

Example 1 (AKML 2021) - Will has a collection of  $12$  identical sticks, and he has decided to paint each stick either red, green, or blue. Given that he wants at least one stick of each color, compute the number of ways for Will to paint his sticks.

Solution to Example 1

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  $\binom{m-1}{s-1}=\binom{11}{2}=55$ .

Open Ended

Example 2 - Roman also has a collection of  $12$  identical sticks and has decided to paint each stick either red, green, or blue. Unlike Will, Roman doesn't care about having all the colors. Compute the number of ways for Roman to paint his sticks.

Solution to Example 2

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  $\binom{m+s-1}{s}=\binom{14}{3}=364$ .