Factoral Notation

COUNTING PRINCIPLES:

This unit is all about counting… without actually counting.

FUNDAMENTAL COUNTING PRINCIPLE

If something can be chosen, or can happen, or be done, in m different ways, and, after that has happened, something else can be chosen in n different ways, then the number of ways of choosing both of them is m · n.

MULTIPLICATION PRINCIPLE:

ü When two events are dependent we multiply their results to find the total number of ways it occurs.

Key Words: “AND”, “BOTH”

ADDITIVE PRINCIPLE:

ü When two events are independent we add their results to find the total number of ways it occurs.

Key Words: “OR”, “EITHER”, “OPPOSITE”

EXAMPLE:

The executive of the Manitoba Association of Mathematics Teachers Association consists of three woman and two men. In how many ways can a president and a secretary can be chosen if:

1. The president is to be a female and the secretary male?

Using Dash Method:

3 and 2 6

President Secretary

2. The president is to be a male and the secretary female?

2 and 3 6

President Secretary

3. The president and secretary are to be of the opposite sex?

6 and 6 12

President Secretary

FACTORIAL NOTATION:

The symbol n! ( read as n factorial) means to multiply all of the positive integers from n consecutively down to 1

The factorial function (symbol: !) just means to multiply a series of descending natural numbers. Examples:

• 4! = 4 × 3 × 2 × 1 = 24
• 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040
• 1! = 1

*n! = n(n-1)(n-2)…(3)(2)(1) , where n is an element of + integer.

EXAMPLE:

ü 24!(n+1)!

22!(n-1)!

ü 24 •23•22!(n+1)(n+0)(n-1)!

22!(n-1)!

ü 552 (n+1)(n+0)

1

ü 552(n+1)n