You can demonstrate possible combinations by actually pushing the buttons on
the lock youve drawn. Make sure students understand the task (to count the number
of combinations and see if it lives up to the billing of thousands) and the rules for
the presses which are:
- Each button may be used at most once.
- Each push may include any number of buttons, from one to five.
Depending on the level of you class, you may want to introduce the two fundamental
rules of counting:
- Count everything once.
- Count nothing more than once.
One way to accomplish rules (1) and (2) is to overcount and adjust.
You can also, as a class, list all the combinations for a 2-button lock. There is a
small enough number to list them all out, but there are enough to get at some
common points of confusion:
Allow the students to spend the majority of the period (at least 30 minutes)
exploring ways to find the number of combinations for a 5-button . You can set the
students to work as individuals or in small groups of 2 or 3. Usually, students will
start out by trying to write down all the combinations for a 5-button lock.
Soon, they will discover they need some systematic way of counting the
As you walk around and observe students work, pay attention to students with
creative, elegant, or clear systemmatic approaches, and be sure to ask them to
share their work at the end of class. Also, be prepared for some groups or
students to be stuck, or to think theyre done quickly. (Often, theyll shout
out 152! as an answer without any explanation.) It can be challenging to
convince them to keep working without saying theyre wrong or giving away the
For these groups of students, it helps to start them on a strategy, especially one
that will help them see any combinations theyve missed. See the Hints document for more details on each of these:
- Think of locks with any number of buttons. Solve the problem for 1, 2, 3,
and 4 button locks. Make a table of combinations, and look for patterns.
- Classify the combinations by how many buttons they use. It is especially
helpful to combine this with (1) above.
- Break down the possible combinations into how many presses they
have. One press means pressing down one or more buttons simultaneously.
So the combination |2|3 4 5|1| is three presses.
- Look at the shape of the combinations. How many look like | * *| * |, or
| * | * *| * |, and so on.
- Write a program in a computer algebra system to generate the actual
combinations. Then, all that is left to be done is count them.
- Make a list of all the combinations on a four-button lock and come up
with a systematic way to extend these to the combinations of the 5-button
- The combinations come in two types: those that use all five buttons and
those that use fewer. Count these separately.
If a group thinks they are done, but have only counted the combinations that use
all five buttons, for example, you might say, Some combinations dont use all 5
buttons. So there are two kinds of combinations, and youve counted all of one kind.
How can you count the others?
If a group is completely stuck, working with a smaller number of buttons is
helpful. You can also encourage them to fill in a table like the one here, and to look
||Number of buttons on lock
|Number of buttons in combination
| 4 || || || || || ||
| 5 || || || || || ||
| Total number of combinations || || || || || ||
You should judge how much your students need in the way of hints or ideas. Try a
table! might be enough for a group that has already been looking at locks with
different numbers of buttons. Other groups may need more structure to get
At the end of the first day, ask several groups to share strategies but no
answers. Ask how they organized their work, what questions they asked
themselves, and so on. If you have one or two strategies that you want to be sure
students see, you can ask groups to present them or you can explain them
yourself. You might also spend some time talking about approaches that are
similar to each other, to get students thinking more broadly about kinds of
approaches. You might decide to encourage students to pursue one of the five main
- A decomposing strategy: How can the combinations on n buttons be
described by combinations on n - 1 buttons? How can combinations with
m pushes be counted by combinations with m - 1 pushes?
- An extending strategy: How can combinations on n - 1 buttons be
extended to combinations on n buttons?
- A direct calculation: Concentrate on the numbers of buttons used.
- A direct calculation: Concentrate on the numbers of pushes used.
- A combinatorial strategy: Find all the different shapes of pushes and use
combinatorics to count the combinations for any particular shape
2 After Day 1
Where you go after the first day of the project is up to you. Here are some
approaches teachers have used in the past:
2.1 Finish Individually After the first day of group work, students are asked to
complete the project individually. Encourage the students to ask questions as they
arise. They usually turn in a rough draft - explaining their approach, their answer,
and any questions that havent been addressed - a week or so after the initial work.
This keeps the students working on the problem and prevents losing momentum from
the first day.
The rough drafts can help you see if there are major misunderstandings or common
mistakes students are making. You can spend some class time, if necessary, going over
these issues when students get comments on their rough drafts. They then finish up
the projects on their own.
2.2 Continuing Group Work You may want to devote a second day of class time
to students working in their groups. At the end of the first day, they may see
connections to other stragies and get ideas for how to continue their work. On the
second day, encourage each group to pick one of the strategies that was discussed and
pursue it to get an answer for the number of combinations of a 5-button
Many groups will arrive at a numerical answer by the end of the first day, or early on
during the second. Often, they do this by a brute-force (though clever) listing of the
combinations. Allow students with different answers to compare methods to see
where under and over counts may have happened.
Once the students agree on the numerical solution, you can ask them to start
making some connections and following up on other questions:
- Can you prove that your answer is right? That is, how can you be sure
you havent missed any combinations or counted any twice?
- How could you generalize the solution? Can you find the number of
combinations for 3-, 4-, and 6-button locks? n-button locks? (see Extension
- One student said they only have to count the combinations that use all five of
the buttons. They double this answer, and claim they have the total number of
combinations. Does that fit your data? Does it work for other numbers of
buttons? Can you explain why it happens? In order to answer these
questions, consider this list of combinations for a three button lock:
| Combinations for a 3-button lock
|Combinations using||Combinations using
| 3 buttons ||less than 3 buttons
| |1|23| || no pushes
| |2|13| || |2|1|
| |3|12| || |12|
| |1|2|3| || |13|
| |1|3|2| || |23|
| |2|3|1| || |3|2|
| |3|1|2| || |3|1|
| |3|2|1| || |1|3|
| |23|1| || |1|2|
| |13|2| || |1|
| |12|3| || |2|
| |2|1|3| || |3|
| |123| || |2|3|
- Finding a one-to-one correspondence between the combinations that use all the
buttons and those that do not use all of the buttons is one way to show that
the two groups have the same amount of elements. Can you find an algorithm
to map each element of the left column to one and exactly one element of the
right column? Would this rule work for a 5-button lock? Would it work for an
n-button lock? How does this information help to answer the above
For students who know about elementary combinatorics and/or students who
have worked on the trains project (see Warm Up Problems), you might ask them
directly how they could use the
formula to help them organize their work and
solve the problem.
For more advanced groups, you can ask them to come up with a closed form or
recursive formula for the number of combinations on an n-button lock. You can also
encourage them to come up with their own extensions to the problem by changing
the question in some way (perhaps changing one or more of the rules about pressing
2.3 Filling in and Coming Back If your students dont yet have the background to
tackle some of the more general solutions, you can interrupt work on the project
(make sure it is well-documented first) for a day or more of background building.
You might want to cover some of these topics:
notation for binomial coefficients and application to the binomial
- summation notation,
in terms of factorials,
- some other work with arithmetic and geometric series,
- the technique of counting objects by putting them in 1-1 correspondence
with better understood sets, and
- counting by extending known sets, for example counting the number of
n-digit numbers using only 1s and 2s by building them up from the single
digits 1 and 2 in a systematic way.
When you come back to the problem, ask students to find a way to use one or more
of these tools as they work on it.