- What is combinatorics? - Mathematics Stack Exchange
In fact,I once tried to define combinatorics in one sentence on Math Overflow this way and was vilified for omitting infinite combinatorics I personally don't consider this kind of mathematics to be combinatorics, but set theory It's a good illustration of what the problems attempting to define combinatorial analysis are
- combinatorics - A comprehensive list of binomial identities . . .
Is there a comprehensive resource listing binomial identities? I am more interested in combinatorial proofs of such identities, but even a list without proofs will do
- combinatorics - What is a combinatorial proof exactly? - Mathematics . . .
Combinatorics is a wide branch in Math, and a proof based on Combinatorial arguments can use many various tools, such as Bijection, Double Counting, Block Walking, et cetera, so a combinatorial proof may involve any (or a combination) of these
- combinatorics - What is $ {n\choose k}$? - Mathematics Stack Exchange
This is the Binomial theorem: $$ (a+b)^n=\sum_ {k=0}^n {n\choose k}a^ {n-k}b^k $$ I do not understand the symbol $ {n\choose k} $ How do I actually compute this? What does this notation mean? Help is
- combinatorics - Why are the formulae related to circular permutations . . .
Circular permutations Consider an arrangement of blue, cyan, green, yellow, red, and magenta beads in a circle For this particular arrangement of the six beads, there are six ways to list the arrangement of the beads in counterclockwise order, depending on whether we start the list with the blue, cyan, green, yellow, red, or magenta bead They correspond to the six linear arrangements shown
- combinatorics - Formula for Combinations With Replacement - Mathematics . . .
If you want a slightly more detailed explanation and exercises I recommend the book Introduction to Combinatorics published by the United Kingdom Mathematics Trust (UKMT) available at their webpage It covers many interesting topics with a problem solving approach to them
- combinatorics - Understanding the stars and bars formula - Mathematics . . .
Imagine we want to put $7$ stars in $3$ bins We can use a visual representation to show how we organise them: $$★ ★ ★ ★ | ★ | ★ ★$$ The bars split the different bins So, according to this graph, $4$ stars are in the first bin, $1$ star is in the second bin and $2$ stars are in the third bin Then, the total number of ways to put $7$ stars in $3$ bins is just the number of ways
- combinatorics - What is Double Counting? - Mathematics Stack Exchange
Can someone please explain what double counting is? I have no idea what it is, and Google search yields results that are too complicated for me to understand at this point in time If there's some
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