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Classifying Combinations: Investigating Undergraduate Students’ Responses to Different Categories of Combination Problems

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Abstract

In this paper we report on a survey designed to test whether or not students differentiated between two different types of problems involving combinations - problems in which combinations are used to count unordered sets of distinct objects (a natural, common way to use combinations), and problems in which combinations are used to count ordered sequences of two (or more) indistinguishable objects (a less obvious application of combinations). We hypothesized that novice students may recognize combinations as appropriate for the first type but not for the second type, and our results support this hypothesis. We briefly discuss the mathematics, share the results, and offer implications and directions for future research.

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Notes

  1. The derivation of the formula for C(n,k) as n!/((n – k)!k!) is not pertinent to the study.

  2. Simple combination problems refer to those that can be solved using a single binomial coefficient; multistep combination problems would require multiple binomial coefficients in the solution (see Table 2 for problems in Survey 1 and the Appendix for Survey 2).

  3. We note that one of the Category I problems (Q17 from Survey 2) and one of the Category II problems (Q19 from Survey 1) had rates that were not consistent with the other problems. We speculate that both problems involved familiar contexts, which might have been a factor for students. This phenomenon underscores the value of having given out multiple surveys with multiple questions so as to avoid any one problem unduly influencing the findings.

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Authors and Affiliations

  1. Department of Mathematics, Oregon State University, 064 Kidder Hall, Corvallis, OR, 97331, USA

    Elise Lockwood

  2. Teachers College, Columbia University, 525 West 120th St., New York, NY, 10027, USA

    Nicholas H. Wasserman & William McGuffey

Authors
  1. Elise Lockwood
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  2. Nicholas H. Wasserman
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  3. William McGuffey
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Corresponding author

Correspondence to Elise Lockwood.

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Appendices

Appendix

Table 5 Survey 2
Full size table

Sets of Outcomes Survey Prompt

Prompt 2: “On the previous page, you entered your solution to different combinatorics problems. On this page, we would like you to expand upon how your solution is related to the “set of outcomes” for a few of the problems. That is, we want you to list some (but not necessarily all) of the outcomes that you are counting. Explain your thinking for your solution and its relation to what is being counted.

For example, for the problem, “If we have four distinct toy cars (Red (R), Blue (B), Green (G), and Yellow (Y)), how many different subsets of 2 of them are there?”, your solution to the problem might have been: C(4,2). On this page, the intent is to expand on how that solution, C(4,2), relates to the set of outcomes. You might write something like, "The set of outcomes includes the following pairs of cars: BR, RG, GY, BG. I used the combination C(4,2) because the outcomes were “pairs” (2) from the 4 different colored toy cars. I did not include RB because this would be the same as BR in this case.”

Overview of Permutations and Combinations

Please read the following page in preparation for the survey questions. You will not be able to return to this page.

Permutations and Combinations

The factorial of a natural number n is the product of all positive integers up to n.

$$ n!=1\ast 2\ast 3\ast \dots \ast \left(n-1\right)\ast n\kern1.00em \mathrm{OR}\kern1.00em n!=n\ast \left(n-1\right)\ast \dots \ast 2\ast 1 $$

A permutation of n distinct objects is an arrangement, or ordering, of the n objects.

For example, if we have four distinct toy cars, and we want to arrange them in a row, we would call such arrangements permutations. Suppose the cars are Red (R), Blue (B), Green (G), and Yellow (Y). Then there are the following 24 permutations of the four cars:

$$ {\displaystyle \begin{array}{llll}\mathrm{RBGY}& \mathrm{GBRY}& \mathrm{RBGY}& \mathrm{YBGR}\\ {}\mathrm{RBYG}& \mathrm{GBYR}& \mathrm{RBYG}& \mathrm{YBRG}\\ {}\mathrm{RGYB}& \mathrm{GRBY}& \mathrm{RGBY}& \mathrm{YGBR}\\ {}\mathrm{RGBY}& \mathrm{GRYB}& \mathrm{RGYB}& \mathrm{YGRB}\\ {}\mathrm{RYBG}& \mathrm{GYBR}& \mathrm{RYBG}& \mathrm{YRBG}\\ {}\mathrm{RYGB}& \mathrm{RYRB}& \mathrm{RYGB}& \mathrm{YRGB}\end{array}} $$

An r-permutation of n distinct objects is an arrangement using r of the n objects. P(n,r) denotes the number of r-permutations. For example, if we wanted to count permutations of size 2 from the four cars, then there are the following 12 2-permutations (or arrangements of 2 of the 4 cars).

$$ {\displaystyle \begin{array}{llll}\mathrm{BR}& \mathrm{GB}& \mathrm{RB}& \mathrm{YB}\\ {}\mathrm{BG}& \mathrm{GR}& \mathrm{RG}& \mathrm{YG}\\ {}\mathrm{BY}& \mathrm{GY}& \mathrm{RY}& \mathrm{YR}\end{array}} $$

An r-combination of n distinct objects is an unordered selection, or subset, of r out of the n objects. C(n,r) denotes the number of r-combinations of a set of n objects. For example, if we have four distinct toy cars, and we want subsets of 2 of them, we have the following 6 2-combinations:

$$ {\displaystyle \begin{array}{lll}\mathrm{BR}& \mathrm{RG}& \mathrm{GY}\\ {}\mathrm{BG}& \mathrm{RY}& \\ {}\mathrm{BY}& & \end{array}} $$

When counting combinations, we only care about the elements in a subset, not in how those elements are arranged.

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Lockwood, E., Wasserman, N.H. & McGuffey, W. Classifying Combinations: Investigating Undergraduate Students’ Responses to Different Categories of Combination Problems. Int. J. Res. Undergrad. Math. Ed. 4, 305–322 (2018). https://doi.org/10.1007/s40753-018-0073-x

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