The concept of factorials had been known since ancient times, but it wasn't until the 17th century that Christian Kramp introduced their current notation, \(n!\)
Factorials arefunctionsin mathematics with the symbol (!), They multiply anumberby every number that precedes it. They can be expressed as \(n!\), where n is the lastnumberof the factorial.
Factorials grow very quickly - so quickly, in fact, that even relatively small factorials can quickly become too large to calculate. For example, the factorial of 20 is over 2 million, while the factorial of 100 is a staggering \(9.3 \cdot 10^{157}\). The largest factorial that can be calculated using a standard calculator is \(69!\), due to limitations in the number of digits that can be displayed.
What is a factorial?
A factorial is simply the product of all positive integers up to a given number. For example, the factorial of 5 is \(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\), or 120.
The factorial rule says the factorial of anynumberis that number times the factorial of the previous number. This can be expressed in a formula as \(n! = n \cdot (n-1)!\) A special case for this is \(0! = 1 \). The symbol n is a whole number, and the exclamation mark represents the expression as factorial.
Factorials can be found in permutations and combinations.
Factorial function
To summarise the above explanation, the factorial function can be expressed as:
\[n! = n \cdot (n-1)!\]
The symbol n is a whole number, and the exclamation mark represents the expression as factorial.
How to calculate factorials
You can go through these steps to find the factorial of a number n.
Let's look at the example of 6!
Write down the sequence of numbers you will multiply by using the factorial formula.
There are six possible combinations that could be made from these three colours in order, and they are:
Blue, red, and yellow --- 1
Blue, yellow, and red --- 2
Yellow, blue, and red --- 3
Yellow, red, and blue --- 4
Red, blue, and yellow --- 5
Red, yellow, and blue --- 6
How many ways can the letters in the word 'forgive' be arranged without repeating them?
Answer: To do a problem like this you count the number of letters in the word 'forgive', and then you find the factorial of it. The number of letters here is 7.
However, notice that \(5!\) can also be found in \(6!\). We could continue with our original method way, but we can also eliminate a chunk of the figures as early as possible. Let's take a look below.
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