System of Equations (aka: Simultaneous Equations): Two or more equations containing common variable(s).

Linear System of Equations: A system of equations in which each equation is linear (first degree).

For any linear system, exactly one of the following will be true:

1.  There is only one solution, the single point of intersection (the system is said to be consistent and independent),
2.  there are infinitely many solutions (consistent and dependent), or
3. there are no solutions (inconsistent).

A linear system with more equations than variables is called overdetermined, and a linear system with more variables than equations is called underdetermined.

Consistent System of Equations: A system of equations that has at least one solution. A consistent and independent system has exactly one solution, the ordered coordinates of the point of intersection. A consistent and dependent system has infinitely many solutions, the equations all graph as the identical line. 

Inconsistent System of Equations: A system of equations which has no solutions. Note: Attempts to solve inconsistent systems typically result in impossible (false) statements such as 0 = 3. The graph is parallel lines. 

Matrix: A rectangular (or square) array of numbers. A matrix is designated by a Capital Letter (matrix A ). Matrices can be written using brackets or parentheses.

Example: 



Element of a Matrix: One of the entries in a matrix. The address of an element is given by listing the row number then the column number. (ALWAYS row, then column.) 
 a₁₂ is the element in the 1^st row, 2nd column. a₂₁ is the element in the 2^nd row, 1st column.
 a₁₃ is -5 and a₃₁ is 4. What is a₃₂?

Coefficient Matrix: The matrix formed by the coefficients in a linear system of equations. If the constants are included in the last column (rightmost), this is called an augmented matrix and usually has a vertical line between the last variable column and the constant column (where the = would be in the equations).

Determinant: A single number obtained from a matrix that reveals a variety of the matrix's properties. The symbol for the determinant of matrix A is |A|. Determinants may be found using expansion by cofactors (but we’re going to use the calculator!).
Note: Although a determinant looks like an absolute value it is not. The determinant of a matrix may be negative or positive.

General counting principle: When you have a number of items with more then one possibility for each item, then the total number of possibilities is the product of the number of possibilities for the first item times number of possibilities for the second item and so on for all items. The fundamental operation of counting is multiplication.

For example, on a test of 5 items with 4 choices for each, you have 4*4*4*4*4, or 4^5 total possiblities. This means that there are 1024 possible ways to arrange the answer key.

Factorial: The product of a given integer and all smaller positive integers. The factorial of n is written n! and is read aloud "n factorial".
Note: By definition, 0! = 1.

Formula: n! = n·(n – 1)·(n – 2) · · · 3·2·1

Example: 6! = 6·5·4·3·2·1 = 720 {Calculator: [ 6 ] [MATH] [PRB] [ 4 ] [ENTER]

Permutation (per-mu-ta-tion): A selection of objects in which the order of the objects matters.

Example: The permutations of the letters in the set {a, b, c} are:
abc        acb 
bac        bca 
cab        cba

Permutation Formula: A formula for the number of possible permutations of k objects from a set of n. This is usually written nPk . Read as, "From n pick k." \*NOTE: Some texts use r instead of k in the formula!!!
 
Formula: 

Example: How many ways can 4 students from a group of 15 be lined up for a photograph? Since the arrangement (order) of how the students are lined-up changes the photo, order of selection DOES matter, so this is a permutation. 

Answer: There are 15P4 possible permutations of 4 students from a group of 15.
 different lineups
{Calculator: [ 1 ] [ 5 ] [MATH] [PRB] [ 2 ] [ 4 ] [ENTER]}

Combination: A selection of objects from a collection. Order is irrelevant. Also, once chosen an object is not replaced in the collection. A combination will always produce a smaller number than a permutation.

Example: A poker hand is a combination of 5 cards from a 52 card deck. This is a combination since the order in which the 5 cards are dealt does not matter.

Combination Formula: A formula for the number of possible combinations of r objects from a set of n objects. This is written in any of the ways shown below.  *NOTE: Some texts use k instead of r in the formula!!! 


All forms are read aloud "n choose r." “n” is the total number of items in the set and “r” is the number to be chosen.
Formula: 
Example: How many different committees of 4 students can be chosen from a group of 15?

Answer: To find the possible combinations of 4 students from a set of 15:

 
There are 1365 different committees.
{Calculator: [ 1 ] [ 5 ] [MATH] [PRB] [ 3 ] [ 4 ] [ENTER]}

Event: A set of possible outcomes resulting from a particular experiment. For example, a possible event when a single six-sided die is rolled is {5, 6}. That is, the roll could be a 5 or a 6.
In general, an event is any subset of a sample space (including the possibility of an the empty set).

Probability: The likelihood of the occurrence of an event. The probability of event A is written P(A). Probabilities are always numbers between 0 and 1, inclusive.

The four basic rules of probability:

1. For any event A, the probability of that event occurring is between 0 and 1. Why? Because the proability of an event occurring is a fraction less than 1. 
2. P(impossible event) = 0. Also written P(empty set) = 0 or P(Ø) = 0.
3. P(sure event) = 1. Also written P(S) = 1, where S is the sample space.
4. P(not A) = 1 – P(A). Also written P(complement of A) = 1 – P(A). This is the probability of any other event than A. See the marble & degrees awarded problems on the review. 

If all outcomes of an experiment are equally likely, then

Retrieved November 17 & 23, 2009, from: http://www.mathwords.com/