Counting Permutations and Combinations

http://www.andrews.edu/~calkins/math/webtexts/prod02.htm

Permutations

Permutation is another name for possible arrangements with SOME items from a given set. It is important to remember that order chosen or position arranged is taken into account. Hence permutations are similar to anagrams. Given below is the necessary equation.
nPr = n! / (n - r)!
where r is the number of items arranged from n elements.

Example: How many ways can you arrange four figurines from a set of seven?
Answer: 7P4=7!/3!=7•6•5•4=840.
Alternate solution: The figurines can be placed as follows: 7 6 5 4 , which is the same as the factorial notation 7!/3!.

http://freeonlinecalculator.net/calculators/discretemathematics/permutations.php

P(8,5) = 8! / (8 - 5)! =
6,720

Combinations

Combinations are arrangements of elements without regard to their order or position.
nCr = n! / (r!(n - r)!)
where r is the number of items taken from n elements.
Note that these numbers are the same as those in Pascal's Triangle, the binomial formula, and the binomial distribution. Those less than about four digits should become very familiar.

Example: You have five places left for stamps in your stamp book and you have eight stamps. How many different ways can you select five?
Answer: 8!/(5!3!) = 8•7•6/(3•2)=56.
Think of putting them in slots, the first has eight choices, the next slot has seven choices and so forth as demonstrated.
8 7 6 5 4

Each combination of choosing 5 out of the 8 has permutations of its own. The five can be arranged in the following ways:
5 4 3 2 1

Thus there are (8!÷3!)÷5! = 8!÷(5!3!)=56 ways to select five of eight, but 6720 ways to arrange five of eight.

C(8,5) = 8! / 5! (8 - 5)! =
56

Unique dice combinations, 6+5+4+3+2+1 = 21, it's like a triangle
for dice with only 4 dots ...
(1,1) (1,2) (1,3) (1,4)
(2,1) (2,2) (2,3) (2,4)
(3,1) (3,2) (3,3) (3,4)
(4,1) (4,2) (4,3) (4,4)

so you can draw a diagonal and you don't count one side of the triangle, so first row has 1, second row has 2, third row has 3, fourth row has 4 so 1+2+3+4 = 10