MCMC (Markov Chain Monte Carlo)

A class of algorithms used to find the probability distribution of an event given the probabilities of causally linked events.

Random walks are taken through the causal chain,

the results of these walks eventually converging on a stationary distribution within an acceptable margin of error.

Combinatorics

nCr : "from n choose r"

n = length of possibility set (ex. 5 letters)
r = length of combination set (ex. pick 2)



Permutation with Repetition
Order matters and elements of the possibility set can be repeated.

nr

	a	b	c	d	e
a	aa	ab	ac	ad	ae
b	ba	bb	bc	bd	be
c	ca	cb	cc	cd	ce
d	da	db	dc	dd	de
e	ea	eb	ec	ed	ee

Permutation without Repetition
Order still matters, but elements cannot be repeated.

n!
(n-r)!

	a	b	c	d	e
a		ab	ac	ad	ae
b	ba		bc	bd	be
c	ca	cb		cd	ce
d	da	db	dc		de
e	ea	eb	ec	ed

Combination without Repetition
Order does not matter, and elements cannot be repeated.

n!
r!(n-r)!

	a	b	c	d	e
a		ab	ac	ad	ae
b			bc	bd	be
c				cd	ce
d					de
e

Combination with Repetition
Order does not matter, but elements can be repeated.

(n+r-1)!
r!(n-1)!

	a	b	c	d	e
a	aa	ab	ac	ad	ae
b		bb	bc	bd	be
c			cc	cd	ce
d				dd	de
e					ee