The Mathematics
of Barcodes

Activity 2: Is the UPC-A Scheme Sufficient?
(Teacher Copy)

Students use the counting principle for combinations, in a context of problem solving, reasoning, and mathematical communication.  Use with the student copy.

### Introduction:

There are 3 people at a party. How many ways can they leave the party? [After making sure that students understand the problem and they come up with solutions, have 3 students act out the situation.] There are 3 x 2 x 1 = 6 ways.  [Discuss the counting principle]

• To use the counting principle;
• To find the number of companies that can be coded into the present Universal Price Code (UPC-A) bar code system.
• To use thecounting principle for combinations to find a new encoding system similar to the UPC-A scheme that will allow a greater number of companies to be encoded.

Part A

1. A total of 12 different sets of clothes.  This problem involving pants and shirts could be solved with a diagram showing that for each pair of pants, each of the four shirts could be worn.

2. There are 3 x 2 x 4 = 24 ways of decorating an apartment before a repeat scheme has to be used.  An organized list might be used for the decorating problem.

3. 100,000 companies.  There are 10 numbers to choose from on each of 5 positions for a total of
10 x 10 x 10 x 10 x 10 or 105 = 100,000.

4. 260,000 license plates.  That is, 26 x 10 x 10 x 10 x 10.

Part B

1. 10,000,000 companies.  That is, the 7-digit code system will accommodate 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10,000,000 companies.

2. Answers will vary.  Students might suggest (a) increasing the number of digits used in the company code, (b) using letters instead of numbers (increasing the number of possibilities for each choice from 10 to 26), or (c) keeping the same basic format as the UPC-A but adding some characters to the numbers 0-9, such as *, #, &, @, +, _.