Author Archives: mzspivey

Pascal Matrices and Binomial Inversion

In this post we’ll look at the relationship between a Pascal matrix, its inverse, and binomial inversion.  It turns out that these are the same concepts viewed from two different angles. The Pascal matrix is the matrix containing Pascal’s triangle … Continue reading

Posted in binomial coefficients, matrices | 2 Comments

Counting Poker Hands

For this post I’m going to go through a classic exercise in combinatorics and probability; namely, proving that the standard ranking of poker hands is correct. First, here are the standard poker hands, in ranked order. Straight flush: Five cards of … Continue reading

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An Expected Value Connection Between Order Statistics from a Discrete and a Continuous Distribution

Years ago, in the course of doing some research on another topic, I ran across the following result relating the expected values of the order statistics from a discrete and a continuous distribution.  I found it rather surprising. Theorem: Fix n, and … Continue reading

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Two Methods for Proving an Alternating Binomial Identity

Recently I gave an optional homework problem in which I asked students to prove the following binomial identity: . (Here, I’m using the Iverson bracket notation in which if P is true and if P is false.) I intended for students to … Continue reading

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A Bonus Question on Convergent Series

Occasionally when teaching the sequences and series material in second-semester calculus I’ve included the following question as a bonus: Question: Suppose  is absolutely convergent.  Does that imply anything about the convergence of ? The answer is that converges.  I’m going … Continue reading

Posted in calculus, sequences and series | 6 Comments

My Experiences on a Post-Election Panel

No mathematics post this month.  Instead, I’m just going to link to an article I published last week in Inside Higher Ed.  In this article I describe my experiences as the conservative voice on a panel held on my campus on November 9.  This was … Continue reading

Posted in campus issues | 1 Comment

Relations That Are Symmetric and Antisymmetric

When I teach relations and their properties, the question of whether a relation can be both symmetric and antisymmetric always seems to arise.  This post addresses that question. First, a reminder of the definitions here: A relation on a set … Continue reading

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