Evaluate the expression.startfraction 9 factorial over 3 factorial endfraction – Beginning with the evaluation of “9 factorial over 3 factorial,” this exploration delves into the intricacies of factorials, their mathematical definition, and their diverse applications across various fields. Factorials, represented by the exclamation mark (!), hold significance in probability, statistics, and combinatorics, making them a cornerstone of mathematical calculations.
This discourse will meticulously break down the expression, guiding you through each step of the evaluation process. By examining alternative methods and exploring real-world applications, we aim to provide a comprehensive understanding of factorials and their practical implications.
Factorial Concepts
A factorial, denoted by the exclamation mark (!), is a mathematical operation that multiplies a positive integer by all the positive integers less than it. For example, 5! is calculated as 5 × 4 × 3 × 2 × 1 = 120.
Factorials have numerous applications in mathematics, including probability, statistics, and combinatorics. They are used to calculate the number of possible arrangements or combinations of a set of objects.
Properties of Factorials, Evaluate the expression.startfraction 9 factorial over 3 factorial endfraction
- 0! = 1
- 1! = 1
- n! = n × (n-1)!
Evaluating the Expression: Evaluate The Expression.startfraction 9 Factorial Over 3 Factorial Endfraction
The expression “9 factorial over 3 factorial” can be broken down as follows:
- 9 factorial = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
- 3 factorial = 3 × 2 × 1
- 9 factorial over 3 factorial = (9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) / (3 × 2 × 1)
Simplifying the expression, we get:
- (9 × 8 × 7 × 6 × 5 × 4) / 1 = 15,120
HTML Table Representation
| Expression | Operation | Result |
|---|---|---|
| 9! | 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 | 362,880 |
| 3! | 3 × 2 × 1 | 6 |
| 9! / 3! | 362,880 / 6 | 60,480 |
Alternative Methods
There are several alternative methods for evaluating the expression “9 factorial over 3 factorial”:
- Using a calculator:Most scientific calculators have a factorial function that can be used to evaluate the expression directly.
- Using a factorial table:A factorial table lists the factorials of all positive integers up to a certain number. This table can be used to look up the values of 9! and 3! and then divide them to get the result.
Real-World Applications
Factorial calculations are used in a variety of real-world applications, including:
- Probability:Factorials are used to calculate the number of possible outcomes in a probability experiment.
- Statistics:Factorials are used to calculate the mean and variance of a binomial distribution.
- Combinatorics:Factorials are used to calculate the number of ways to arrange or select objects from a set.
Essential Questionnaire
What is the mathematical definition of a factorial?
A factorial, denoted by the exclamation mark (!), is a mathematical operation that calculates the product of all positive integers up to a given number. For instance, 5! is calculated as 5 x 4 x 3 x 2 x 1, resulting in 120.
How do you evaluate the expression “9 factorial over 3 factorial”?
To evaluate the expression, we first calculate 9 factorial and 3 factorial separately. Then, we divide 9 factorial by 3 factorial to obtain the final result.
