# An Analysis of Statements of the Multiplication Principle in Combinatorics, Discrete, and Finite Mathematics Textbooks

## Abstract

The multiplication principle serves as a cornerstone in enumerative combinatorics. The principle underpins many basic counting formulas and provides students with a critical element of combinatorial justification. Given its importance, the way in which it is presented in textbooks is surprisingly varied. In this paper, we analyze a number of university textbooks in order to explore and identify several key aspects of the various statements of the principle. We characterize the nature of the variety of statements by identifying structural, operational, and bridge statements, and we highlight the respective statements’ mathematical and pedagogical implications. We conclude by indicating several directions for future research.

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## Notes

1. 1.

We follow a number of authors in using the term “multiplication principle” throughout the paper, even though the textbooks we surveyed had many different names for it.

2. 2.

She clarifies that “the word ‘concept’ (sometimes replaced by ‘notion’) will be mentioned whenever a mathematical idea is concerned in its ‘official’ form” (p. 3), whereas a “conception” refers to “the whole cluster of internal representations and associations evoked by the concept” (p. 3).

3. 3.

We did not include universities outside of the United States in order to limit the scope and because we did not feel equipped to linguistically analyze textbooks in other languages. We also did not examine probability textbooks, primarily because we suspect that reasoning about multiplication in probability contexts may fundamentally differ from strictly combinatorial contexts, and thus the inclusion of such texts would require further analysis and would complicate our narrative. Each of these ideas represents a potential direction for future research.

4. 4.

The expression 9 × 13 + 3 × 12 = 153 results from a case breakdown that attends to the issue of dependence, where the first case is if a non-heart face card is drawn first, and the second case is if a heart face card is drawn first.

5. 5.

A classic example of this is the relationship between the formulas for permutations and combinations, in which each selection of k objects from n distinct objects is actually equivalent to k! arrangements of k objects from n objects. To find the number of combinations we can count permutations and divide by k!

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Lockwood, E., Reed, Z. & Caughman, J.S. An Analysis of Statements of the Multiplication Principle in Combinatorics, Discrete, and Finite Mathematics Textbooks. Int. J. Res. Undergrad. Math. Ed. 3, 381–416 (2017). https://doi.org/10.1007/s40753-016-0045-y