[download RTF (text) file which includes this text and student worksheet on second page]
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This is a group project: I randomly assign the students to groups of 3 or so (using a deck of cards), then circulate through the room answering questions.
You'll notice that the missing formula is not explicitly derived in full generality. I address that in a subsequent lecture that follows up on the project. The formula, in the nCr notation, is
(n + r - 1)Cr = (n + r - 1)C(n - 1)
(whether you want to think about placing dots / scoops or bars / separators, respectively).
This formula is addressed in most discrete math, combinatorics, and probability texts, but I find it accessible to first year finite math students. The table of counting formulas helps motivate the question and solidify understanding of previous material (which is also used to fill in the question marks).
Feel free to modify this in any way that helps (I'm sure I'll tinker with it again next time). And please let me know if and how you end up using it.
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This material is based upon work supported by the National Science Foundation under Grant No. 0314808.