Counting, Multiplication Principle, Pigeonhole Principle and Reasoning for Middle Schoolers and Beyond

UPDATE: SEE THE NEW VIDEO BELOW EXPLAINING THE PROBLEMS IN THIS POST. PLS SUBSCRIBE TO THE NEW MathNotationsVids Channel and share your comments and ratings!




The following video is available on my new MathNotations Videos Channel.


This particular video is a 10 minute discussion of developing the Multiplication Principle of Counting. It is designed more for the instructor than the student although it may be helpful in clarifying this important concept. The focus is on using multiple representations to reach the widest variety of learning styles. It is appropriate for any teacher of mathematics but particularly for the middle school teacher or those who work with students who struggle with math concepts.


After watching the video (or skip it if you wish) scroll down to the two problems below. These are more sophisticated than the one in the video and they require application of other concepts as well. I believe they are appropriate for 8th graders through high school. A full investigation with questions is provided for each problem. Feel free to edit them to your own tastes or as needed for your students.

Problem I

Mr. M told his Period I 8th grade math class about the following imaginary scenario...

Before the first day of school, Mr. Serling noticed that the names of the 26 students in his 1st period class had an unusual property. All of their initials (First Initial, Last Initial) came from the letters A, B, C, D and E. Furthermore, some had duplicate initials like B.B.

Part (a)
He now asked his actual class to make a conjecture:

Do you think it's possible that all 26 students in this imaginary class could have different initials (from each other)?  Write down your "initial" prediction (Y or N) on a slip of paper and fold it over. 

Part (b) Ok, now that you've made a conjecture, get into your learning groups of 4 and individually make a list of all possible sets of initials using the letters A, B, C, D and E with repetitions like "B.B." allowed as I explained before. Make sure your lists agree - edit as needed. Are your lists easy to compare? Why or why not?

HOW MANY DIFFERENT SETS OF INITIALS DID YOUR GROUP AGREE ON? ________

Part (c) Show your predictions to your partners and, in pairs, explain your reasoning why you would stay with your original prediction or change. Then write your reasoning as follows:

I believe that it is/is not possible for the 26 students to have different initials because ___________________________________.


At this point, Mr. M reviewed the Multiplication Principle of Counting (see the video above).

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The following problem may be assigned for classwork or homework after Problem I has been discussed in class. You could also use it as an assessment.


Problem II
Mr. M decides to assign to each student in his 5 classes a unique code consisting of up to 5 colors in sequence. He has a total of 129 students and the codes will use only the colors Red, Yellow, Green, Blue and Purple. Mr. M explains that codes may have repeated colors (like GGG or GYG) and RYG is a different code from YGR.

Will Mr. M run out of different codes for his 129 students? Explain your answer carefully, using a method similar to Problem I. 

Comment: I haven't mentioned how the Pigeonhole Principle can be applied to these two problems. I'll leave it to my astute readers to comment on that!




Ok, here's another video explaining the two problems  above. I hope you will subscribe to my new channel on YouTube, MathNotationsVids. 



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Note: I've been asked why I'm using these signatures on my posts, particularly the 2nd one. Well...
"It's my party and I'll try what I want to!"
(Apologies to Lesley!)

I'm sure some of my devoted readers can figure out why I included Schopenhauer's quote and the 2nd one is really all about education, isn't it?


"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)

You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught.
--from South Pacific