SAT Factorial Definition, Notation, Formula, Examples

Let's dive into the concept of factorials, a super important math function! Factorials come up all the time in areas like probability, combinations, and even sequences. You can think of a factorial as a way of multiplying a number by all the numbers below it, all the way down to 1. So, for example, 3! (which is read as "three factorial") equals 3 × 2 × 1, which gives you 6.

Why does this matter for exams like the SAT, ACT, or AP exams? Factorials pop up in probability problems, permutations, and combinations, so knowing how to quickly calculate them is a big help. They also show up in things like counting problems, sequences, and series. So, whether you're working through a tricky math question or a word problem, understanding factorials will give you an edge on test day!

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Defining Factorial

In Mathematics, factorial is a simple yet powerful concept. Factorials are just products, indicated by an exclamation mark. Factorial is the multiplication of a number with all the natural numbers that are less than it. Let's delve deeper into the definition, formula, and examples of factorial.

Notation of Factorial

Factorial is represented by the symbol “ n! ”. It is the multiplication of all positive integers, say “n”, that are smaller than or equal to n.

Formula for Factorial

The formula to calculate the factorial of a number is as follows:

n! = n × (n-1) × (n-2) × (n-3) × ….× 3 × 2 × 1

For an integer n ≥ 1, the factorial representation in terms of pi product notation is:

\(\begin{array}{l}n! = \prod_{i=1}^{n}i\end{array} \)

From the above formulas, the recurrence relation for the factorial of a number is defined as the product of the factorial number and factorial of that number minus 1. It is given by:

n! = n. (n-1) !

Calculating Factorial of a Number

To calculate the factorial of any given number, you simply substitute the value for n in the above formula. The expansion of the formula gives the numbers to be multiplied together to get the factorial of the number.

Computing Factorial of 10

For example, the factorial of 10 can be calculated as follows:

10! = 10. 9 !

10! = 10 (9 × 8 × 7 × 6 × 5× 4 × 3 × 2 × 1)

10! = 10 (362,880)

10! = 3,628,800

Therefore, the factorial of 10 is 3,628,800

The factorial function is widely used in various fields of Mathematics such as algebra, permutation and combination , and mathematical analysis. Its main application is to count the possible distinct arrangements of “n” objects.

For instance, the number of ways in which 4 persons can be seated in a row can be found using the factorial. That means, the factorial of 4 gives the required number of ways, i.e. 4! = 4 × 3 × 2 × 1 = 24. Hence, 4 persons can be seated in a row in 24 ways.

Factorial Table for Numbers 1 to 10

Here is a list of factorial values for numbers 1 to 10:

n	Factorial of a Number (n!)	Expansion	Value
1	1!	1	1
2	2!	2 × 1	2
3	3!	3 × 2 × 1	6
4	4!	4 × 3 × 2 × 1	24
5	5!	5 × 4 × 3 × 2 × 1	120
6	6!	6 × 5 × 4 × 3 × 2 × 1	720
7	7!	7 × 6 × 5 × 4 × 3 × 2 × 1	5,040
8	8!	8 × 7 × 6 × 5 × 4 × 3 × 2 × 1	40,320
9	9!	9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1	362,880
10	10!	10 × 9 ×8 × 7 × 6 × 5 ×4 × 3 × 2 × 1	3,628,800

Understanding Sub Factorial

Sub factorial, represented by “!n”, is a mathematical term defined as the number of rearrangements of n objects. It refers to the number of permutations of n objects where no object is in its original position. The formula to calculate the sub-factorial of a number is as follows:

\(\begin{array}{l}!n = n!\sum_{k=0}^{n}\frac{(-1)^{k}}{k!}\end{array} \)

Calculating Factorial of 5

Calculating the factorial of 5 is straightforward. You can find it by using the formula and expanding the numbers. Let's see how it's done.

We know that,

n! = 1 × 2 × 3 …… × n

Factorial of 5 can be calculated as follows:

5! = 1 × 2 × 3 × 4 × 5

5! = 120

Therefore, the factorial of 5 equals 120.

Factorial Examples

Example 1:

What is the factorial of 6?

Solution:

We know that the factorial formula is

n! = n × (n – 1) × (n – 2) × (n – 3) × ….× 3 × 2 × 1

So the factorial of 6 is

6! = 6 × (6 -1) × (6 – 2) × (6 – 3) × (6 – 4) × 1

6! = 6 × 5 × 4 × 3 × 2 ×1

6! = 720

Therefore, the factorial of 6 equals 720.

Example 2:

What is the factorial of 0?

Solution:

The factorial of 0 equals 1

i.e., 0 ! = 1

The result of multiplying no factors is a nullary product. It means that the convention equals the multiplicative identity.

Practice Problems

Practice the problems given below to better understand the concept.

Evaluate 8! – 6!.
What is the value of 14!/(12! 2!)
If (1/7!) = (x/9!) – (1/8!), then what is the value of x?
Is 5! + 6! = 11!?

Conclusion

To wrap it up, factorials are a key concept in math, especially when dealing with problems related to probability, permutations, and combinations. They help you count and arrange things in many different ways, which is super useful for exams like the SAT, ACT, or AP tests. Understanding how to calculate and use factorials can give you a big advantage on test day and in solving math problems more efficiently!

Frequently Asked Questions

What is a factorial of 10?

The value of factorial of 10 is 3628800, i.e. 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3628800.

What is the meaning of 5 factorial?

The meaning of 5 factorial is that we need to multiply the numbers from 1 to 5. That means, 5! = 5 × 4 × 3 × 2 × 1 = 120.

What is the symbol of factorial?

The factorial function is a mathematical formula represented by an exclamation mark “!”. For example, the factorial of 8 can be represented as 8! and it is read as eight factorial.