1. Meaning of Factorial (n!)
Factorial is a way of multiplying a number with all the positive whole numbers before it. It is written using the symbol !, called the factorial sign.
When we write \(n!\), it means we multiply all numbers from \(1\) to \(n\). This idea is used a lot in permutations and combinations.
1.1. Definition of Factorial for Natural Numbers
For any natural number \(n \ge 1\):
\( n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1. \)
Example: \(5! = 5 \times 4 \times 3 \times 2 \times 1.\)
1.2. Simple Numerical Values of n!
Some common factorial values are:
- \(1! = 1\)
- \(2! = 2\)
- \(3! = 6\)
- \(4! = 24\)
- \(5! = 120\)
These values grow fast as \(n\) increases.
2. Special Cases: 0! and 1!
Two important special cases in factorials are \(0!\) and \(1!\). Both are equal to 1. These values help formulas work smoothly in permutations and combinations.
2.1. Why 0! Is Defined as 1
\(0!\) is defined as 1 so that counting formulas remain consistent. For example, in permutations we use:
\( nP0 = \frac{n!}{(n-0)!} = 1 \)
So defining \(0! = 1\) fits naturally.
2.2. Understanding 1!
\(1! = 1\) because there is only one way to arrange a single object. It also fits the factorial definition:
\(1! = 1\)
3. Basic Properties of Factorial
Factorials have some simple properties that help in simplifying expressions. These properties also form the base for many P&C formulas.
3.1. Recursive Relation n! = n × (n−1)!
Factorials follow a "step-down" pattern:
\( n! = n \times (n-1)! \)
Example: \(5! = 5 \times 4!\)
3.2. Product Form of Factorial
Factorial can be written as a long product:
\( n! = 1 \times 2 \times 3 \times \dots \times n \)
This form helps when expanding terms.
3.3. Growth of Factorials (Intuitive Idea)
Factorials grow very quickly. Even small increases in \(n\) create large values.
Example: \(6! = 720\), \(7! = 5040\), \(8! = 40320\).
4. Simplifying Expressions Involving Factorials
Many expressions in permutations and combinations contain factorials. Knowing how to simplify them makes calculations easier.
4.1. Cancelling Factorials in Fractions
Often we see expressions like:
\(\frac{n!}{(n-1)!} = n\)
Most terms cancel, leaving only the top number.
4.2. Expanding Factorials to Simplify
To simplify expressions, we sometimes expand factorials only partly.
Example:
\(\frac{7!}{5!} = \frac{7 \times 6 \times 5!}{5!} = 7 \times 6 = 42\)
5. Factorials in Counting Problems
Factorials appear naturally in problems where the order of objects matters.
5.1. Arrangements of Distinct Objects
The number of ways to arrange \(n\) different objects in a row is:
\( n! \)
Example: 4 objects can be arranged in \(4! = 24\) ways.
5.2. Connection with Permutations and Combinations
Permutation and combination formulas use factorials.
Examples:
- \( ^nP_r = \frac{n!}{(n-r)!} \)
- \( ^nC_r = \frac{n!}{r!(n-r)!} \)
These formulas work because factorials count step-by-step arrangements.