Pascal's Triangle (see additional factorial and binomial coefficient information below) ...

1
1   1
1   2   1
1   3   3   1
1   4   6   4   1
1   5   10   10   5   1
1   6   15   20   15   6   1
1   7   21   35   35   21   7   1
1   8   28   56   70   56   28   8   1
1   9   36   84   126   126   84   36   9   1
1   10   45   120   210   252   210   120   45   10   1

Factorials

N! = N * (N-1) * (N-2) * (N-3) * ... * 3 * 2 * 1   :where N is a positive integer. In many math textbooks, factorial is introduced as the number of ways that an original set can be ordered. For example, if an original set contains 3 items {A, B, C}, then the number of ways that you can order these three items (set elements) is 3! = 3*2*1 = 6 ({A,B,C}, {A,C,B}, {B,A,C}, {B,C,A}, {C,A,B}, {C,B,A}). Note: 0! is defined to equal 1.

Number
Factorial
 ... ... ... ... ... Number
Factorial
0
1
11
39,916,800
1
1
12
479,001,600
2
2
13
6,227,020,800
3
6
14
87,178,291,200
4
24
15
1,307,674,368,000
5
120
16
20,922,789,888,000
6
720
17
355,687,428,096,000
7
5040
18
6,402,373,705,728,000
8
40,320
19
121,645,100,408,832,000
9
362,880
20
2,432,902,008,176,640,000
10
3,628,800
21
BIG!!!!

Permutations: If "M" denotes the number of permutations of "N" things taken "P" at a time, then M = N * (N-1) * (N-2) * ... * (N-P+1) = N!/P! (where "!" represents factorial).

Binomial Coefficient (Combinations)

The Binomial Coefficient, also sometimes referenced as the number of uniquely different (distinctly different) ways that you can select a subset of size "K" form an original set of size "N", is given by the following definition:

For example, if asked how many different ways that one could select a subset of size 3 from an original set of size 5, the answer is (5!)/(2! 3!) = 10. Original set of size five:{A,B,C,D,E} and distinctly different subsets of size 3: {A,B,C}, {A,B,D}, {A,B,E}, {A,C,D}, {A,C,E}, {A,D,E}, {B,C,D}, {B,D,E}, {B,C,E}, and {C,D,E}. Note: Duplicate listings that are NOT distinct ... for example {C,A,B}... are NOT repeated nor listed.

It turns out that the numbers generated by the Binomial Coefficient are identical to those in Pascal's Triangle ...
K=0 K=1 K=2 K=3 K=4 K=5 K=6
N=0 1 NA NA NA NA NA NA
N=1 1 1 NA NA NA NA NA
N=2 1 2 1 NA NA NA NA
N=3 1 3 3 1 NA NA NA
N=4 1 4 6 4 1 NA NA
N=5 1 5 10 10 5 1 NA
N=6 1 6 15 20 15 6 1

where "N" is the row number in Pascal's Triangle (starting a count with row 0), and "K" is the position in the Pascal's Triangle row (starting a count with position 0).

Binomial Expansion

Using either Pascal's Triangle or the Binomial Coefficient allows an individual to rapidly expand a binomial in Algebra. Basically ... the expansion is a mix of three patterns: the binomial coefficient numbers (form Pascal's Triangle), a descending power sequence of the first term in the binomial, and an ascending power sequence of the second term in the binomial. The following illustrates the pattern:

( A + B )0  =  1

( A + B )1  =  1A  +  1B

( A + B )2  = 1A2  +  2AB  +  1B2

( A + B )3  =  1A3  +  3A2B  +  3AB2  +  1B3

( A + B)4  =  1A4  +  4A3B  +  6A2B2  +  4AB3  +  1B4
 


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