Chapter 21 Balls in Boxes Part 1
Answer the following questions by turning each problem into a balls in boxes problem. (What are the balls? What are the boxes?)
Exercises Practice Problems
1. Puppies!
How many ways are there to give four doggie treats to six puppies, with no more than one doggie treat to each puppy? Solution.
This time, the distinct boxes are the puppies. The identical balls are the doggie treats. Boxes can be empty and repetition is not allowed. So the answer is:
\begin{equation*} {6 \choose 4} \end{equation*}How many ways are there to give four doggie treats to six puppies when you can give multiple treats to the same puppy? Solution.
This time, the distinct boxes are the puppies. The identical balls are the doggie treats. Boxes can be empty and repetition is allowed. So the answer is:
\begin{equation*} {6+4-1 \choose 6} = {9 \choose 6} \end{equation*}
2. Stars.
How many ways are there to put a star sticker (they come in gold, silver, red, green, and blue) on every student's paper in a class of 30 students? Solution.
We have to make 30 decisions, and we have 5 options for each. The number of ways to do this is
Here the colors of the stars are the 5 distinct boxes. The students are 30 distinct balls. Repetition is allowed and boxes can be empty.
3. Fruit Selection.
How many ways are there to buy 10 pieces of fruit, when you can choose between four kinds: apples, bananas, oranges and pears? (Note: if we buy 2 apples, we treat the apples as "the same.") Solution.
The distinct boxes are the 4 types of fruit. We place 10 identical balls into these boxes: the number of balls in a box is the number of pieces of that type of fruit. Boxes can be empty and repetition is allowed. The number of ways to do this is
4. An Extra Slice of Pie.
How many ways are there to give 4 slices of apple pie and 6 slices of pumpkin pie to nine hungry diners, where each diner gets exactly one slice of pie. (So there is one slice left over!) Solution.
There are two cases, depending on which type of pie is left over: pumpkin or apple.
In the first term, we pick the 4 people who will get apple pie. The remaining 5 get pumpkin pie. The left over slice is apple
In the second term, we pick the 6 people who will get apple pie. The remaining 3 get pumpkin pie. The left over slice is pumpkin.
5. Spooky Minifigs.
How many ways are there to arrange 6 identical ghost figurines and 9 identical zombie figurines on a shelf? Solution.
We must choose the 6 spots for the ghosts. The zombies go in the other spots. The number of ways is
\begin{equation*} {15 \choose 9} \end{equation*}Here the 15 spots (from left to right) are the distinct boxes. We use 6 identical balls to indicate the ghosts. Boxes can be empty and repetition is not allowed.
How many ways are there to arrange 6 distinct ghost figurines and 9 distinct zombie figurines on a shelf? Solution.
In this case, all 15 figurines are distinct, and we must line them up. The number of ways is
\begin{equation*} 15! \end{equation*}Here the distinct boxes are the 15 possible positions. The 15 distinct balls are the figurines. Repetition is not allowed and boxes cannot be empty.
6. Toys for Tots.
How many ways are there to give three distinct toys to three out of five children, such that no child gets more than one toy? Solution.
The distinct boxes are the 5 children. The distinct balls are the 3 toys. Repetition is not allowed and empty boxes are okay. The number of ways is