n factorial, denoted as n!, is the product of all positive integers from 1 to n. It is defined as:
n! = n × (n − 1) × (n − 2) × ... × 3 × 2 × 1
For example:
5! = 5 × 4 × 3 × 2 × 1 = 120
By convention, 0! = 1, which is known as the empty product and is consistent with combinatorial interpretations (e.g., there is exactly one way to arrange zero objects).
Key Properties and Applications
Recursive Definition: n! = n × (n − 1), with 0! = 1 as the base case.
Permutations: The number of ways to arrange n distinct objects in a sequence is n!.
Combinations: The binomial coefficient (n choose k) is calculated as n! / (k! × (n − k)!).
Mathematical Analysis: Factorials appear in Taylor series expansions, such as e^x = Σ (x^n / n!).
Growth Rate: n! grows faster than exponential functions. For large n, Stirling’s approximation provides an estimate: n! ≈ √(2πn) × (n/e)^n.
Historical and Symbolic Notes
The notation n! was introduced by Christian Kramp in 1808.
Factorials have ancient roots in Indian mathematics (Jain literature), Hebrew mysticism (Sefer Yetzirah), and Arab grammar.
The factorial function can be extended to non-integer and complex numbers using the gamma function: n! = Γ(n + 1), which is undefined for negative integers.
For quick computation, use a factorial calculator (e.g., CalculatorSoup or RapidTables).
Factorials - Explain like im a two year old
why does n factorial work for determining the numebr of arranegemtns?
Anyone know how to make a factorial n! In flowgorithm?
How is log (n factorial) equal to omega (n log n)?
log (n!) = log [n * (n - 1) * (n - 2) * ... * 1], n things are being multiplied.
>= log [n * (n - 1) * ... * (n / 2)], n/2 things are being multiplied here, so I'm just dropping the second half.
>= log [(n / 2)^(n / 2)], because each one is greater than or equal to n / 2.
= n/2 log (n / 2). More on reddit.com Videos
The factorial definition is given on wikipedia:
https://en.wikipedia.org/wiki/Factorial#Definition
as n! = 1.2.3.....
But when you put n=1 into it, it will always give you zero, because of (n-1) term, so I don't understand it? Where am I abusing the maths here (I know I am making a mistake just dont understand where or why)?
EDIT: Thanks for the help, Giannie and gkikola have given me the answer which helped. Thanks everyone who posted as well!
we just started learning perms and combs at school, and we learnt that if there are n distinct objects, then the number of ways it can be arranged in a line is in 'n!' ways; i tried to understand why because our textbook never really explained the underlying reason for why this works and how this makes sense practically.
what i was thinking is that we visualise the scenario of items as a tree diagram; each stage represents the first item, and the number of branches represent the number of items that are available.
Say we has letters A, B and C (and we want to determine the number of arrangements of a word with these three letters); the first letter can either be A, B or C, so the first stage has three branches open; then we have two letters left. At each of the three branches they branch out further into two branches; and then branch out once more (only one branch). What I observed was that at the first stage of picking the first letter we had three branches, and then they branched out furhter into 2 each, so we obtain 3x2 = 6 combinations (in the process); then they branch out once more, so we get 3x2x1 = 6 combinations = 3!.
could someone explain if my understanding for why the n factorial works like this? I'd other appreciate any other ways that I can visualise this! thanks!