Some of my 16 year old students hanker after the formula for the # of k-permutations of n objects, with x types, where $r_1, ⋯, r_x$ = the number of each type of object. This is more generalized than this question.

What books accessibly teach this formula? What books gently expound — fill in all gaps and steps in — this answer by Prof. Suresh Venkatasubramanian?

The book doesn't have to be written for 16 year olds. You can recommend books for undergraduates, but they must be readable and easygoing.

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    $\begingroup$ Rather than a textbook, which at the high school level would be much broader in scope (covering many topics that would be taught throughout the high school year the book is used for) and thus would probably only cover something this specialized rather tangentially if at all, I recommend something intended as supplementary reading for better students, such as Mathematics of Choice. How to Count Without Counting by Ivan Niven (1965; .pdf file). $\endgroup$ Commented Jan 9, 2022 at 7:03
  • $\begingroup$ @DaveLRenfro thanks! I edited my answer. I didn't mean to limit question to textbooks for high school. $\endgroup$
    – user155
    Commented Jan 9, 2022 at 7:46
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    $\begingroup$ Best derivation I've come around is "The Tao of BOOKKEEPER". $\endgroup$
    – vonbrand
    Commented Jan 13, 2022 at 17:30
  • $\begingroup$ I think Hart College Algebra, chapter on permutations and combinations, has right coverage and easiness. I'm on my phone so can't cut and paste, but archive.org has a pdf of the 1926 edition. $\endgroup$
    – guest
    Commented Dec 2, 2022 at 17:39
  • $\begingroup$ @DaveLRenfro,vonbrand The question indicates (check the links, especially the mathoverflow one) that $k$ is allowed to be less than $\sum r_i$. I was not able to find a discussion of this much harder version of the problem in Niven, Tao of BOOKKEEPER, or Hart. If it's in one of those and I missed it, it might be helpful to provide page references. $\endgroup$ Commented Jan 1, 2023 at 22:41

2 Answers 2


Another recommendation ist Vilenkin: Conbinatorics, see https://www.isinj.com/mt-usamo/Combinatorics%20Vilenkin%20N.Ia.%20(1971).pdf

A good book for undergraduates is

L. Lovász, J. Pelikán, K. Vesztergombi:Discrete Mathematics, elementary and beyond. Springer, 2003, ISBN: 978-0-387-95585-8, https://doi.org/10.1007/b97469


The lecture notes from Xinwei Yu's Finite Mathematics course at the University of Alberta contain a discussion that may be at the right level. See Generating Functions II, Example 5. This example is slightly more general than your problem, but describes the necessary techniques in detail. Exercise 2 is exactly your problem.


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