# Monthly Archives: May 2011

## All models are wrong!

The actual quote, attributed to George Box, according to here is all models are wrong but some are useful How true.

## Number Theory by

Pommersheim, Marks and Flapan. The full subtitle of the book is “A Lively Introduction with Proofs, Applications, and Stories.” I have to admit I only browsed through a couple of pages of the book but it already lives up to … Continue reading

Posted in Books, Number Theory | 1 Comment

## A dicey past year exam question

A die consists of six faces with each face representing precisely one of the numbers 1, 2, 3, 4, 5, 6. Suppose that n such dice are rolled for some positive integer n. The number of the upper face of … Continue reading

## Modelling with exponential generating functions

Let $a_n$ be the number of ways to distribute n distinct objects to four distinct boxes, such that the total objects is even. Find the exponential generating function. Either all four boxes have even number of objects, or exactly … Continue reading

## Binomial identity and probability

The identity $\displaystyle \sum k \binom{n}{k} = n 2^{n-1}$ is pretty standard, and one can prove it algebraically by cancelling the k in the sum with the binomial coefficient and then using the binomial theorem summation or a combinatorial … Continue reading

## Double Factorial

Using the double factorial notation to denote the following $\displaystyle n!! = \prod_{i=0}^{\lfloor \frac{n-1}{2} \rfloor} (n-2i)$ seems pretty standard. (See Wolfram and Wiki.) So $4!! = 4 \times 2 = 8$ but $(4!)! = 24!$. … Continue reading

## Pascal’s triangle

Perhaps the most famous triangle of all. Take your calculator, and compute $11, 11^2, 11^3, 11^4$ … cute! Can you explain why? It’s so famous that there’s lots of information on the web about it. Named after Pascal but … Continue reading

## Zeta(5) is irrational ?

The answer is that it probably is but mathematicians do not yet know how to prove it. A paper has been put up in arXiv (dated 4 May) that claims to have used very elementary methods to prove that $\zeta(5)$ … Continue reading