What is Factorial: Definition and 162 Discussions

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n:

For example,

The value of 0! is 1, according to the convention for an empty product.The factorial operation is encountered in many areas of mathematics, notably in combinatorics, algebra, and mathematical analysis. Its most basic use counts the possible distinct sequences – the permutations – of n distinct objects: there are n!.
The factorial function can also be extended to non-integer arguments while retaining its most important properties by defining x! = Γ(x + 1), where Γ is the gamma function; this is undefined when x is a negative integer.

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  1. Hill

    A "spiral" in the Complex plane

    I understand that the "spiral" converges to 1+i-1/2-i/3!+1/4!+i/5!-1/6!-i/7!... . It splits into two: one for Re, 1-1/2+1/4!-1/6!..., and the other for Im, 1-1/3!+1/5!-1/7!... . Any hints on how to compute them?

    Finding the Last Non-Zero Digit of a Repeated Factorial Expression

    Without using computer programs, can we find the last non-zero digit of $$(\dots((2018\underset{! \text{ occurs }1009\text{ times}}{\underbrace{!)!)!\dots)!}}$$? What I know is that the last non-zero digit of ##2018!## is ##4##, but I do not know what to do with that ##4##. Is it useful that...
  3. Kyuubi

    B Cool fact about number of digits in n!

    This may have already been found by many people but I discovered the pattern on my own out of curiosity with some coding. There are only 4 natural numbers whose factorial contains the same number of digits as the number itself. That is to say n = digits_in(n!). The trivial case is obviously...
  4. homeworkhelpls

    How do we get from one step to another in these factorial equations?

    this is the answer, but i don't get why k factorial multiplies the bracket, what i did was k factorial divided by the bracket
  5. S

    MHB Find A,B,C for Factorial Equation

    find A,B,C such that: ABC= A!+B!+C!
  6. R

    Stirling's Approximation for a factorial raised to a power

    Using log identities: ##log((\alpha - 1)!^2) = 2(log(\alpha - 1)!)## Then apply Stirling's Approximation ##(2[(\alpha - 1)log(\alpha - 1) - (\alpha - 1)## ## = 2(\alpha -1)log(\alpha -1) - 2\alpha+2## Is this correct? I can't find a way to check this computationally.
  7. U

    I Prove series identity (Alternating reciprocal factorial sum)

    This alternating series indentity with ascending and descending reciprocal factorials has me stumped. \frac{1}{k! \, n!} + \frac{-1}{(k+1)! \, (n-1)!} + \frac{1}{(k+2)! \, (n-2)!} \cdots \frac{(-1)^n}{(k+n)! \, (0)!} = \frac{1}{(k-1)! \, n! \, (k+n)} Or more compactly, \sum_{r=0}^{n} (...
  8. anemone

    MHB What are the Positive Integer Solutions to the Factorial Equation?

    Determine all positive integers $a,\,b$ and $c$ that satisfy equation $(a+b)!=4(b+c)!+18(a+c)!$.
  9. Saracen Rue

    I Define the double factorial as being a continous, non-hybrid function

    The double factorial, ##n!## (not to be confused with ##(n!)!##), can be defined for positive integer values like so: $$n!=n(n−2)(n−4)(n−6)...(n-a)$$ Where ##(n−a)=1## if ##n## is odd or ##(n−a)=2## if ##n## is even. Additionally, the definition of the double factorial extends such that...
  10. anemone

    MHB Inequality involves radical, square and factorial expression 3√{x}+2y+1z^2⩽ 13

    If $x^2+y^2+z^2+xyz=4$ and that $x,\,y,\,x\ge 0$, prove $3!\sqrt{x}+2!y+1!z^2\le 13$.
  11. hilbert2

    A Analytical function from numerable point set

    Sometimes there are functions that are initially defined for only integer values of the argument, but can be extended to functions of real variable by some obvious way. An example of this is the factorial ##n!## which is extended to a gamma function by a convenient integral definition. So, if I...
  12. Vital

    I Understanding part of the multinomial formula

    Hello. I am trying to decipher the formula, making sure I understand what exactly is going on in each part of the expression. I will be grateful for your guidance, corrections and help. Below I show the formula and the example for only 3 possible outcomes (in general it would be k) \(p =...
  13. physea

    2^k factorial experiment design

    Hello! In this webpage: https://onlinecourses.science.psu.edu/stat503/node/35 it describes the 2^k factorial experiment design. I understand that k is the number of factors that we are investigating (in this case two, a and b), 2 are the levels of each factor (+/-) and 2^k=4 is the number of...
  14. J

    Proof for Γ(p+1/2) using Double Factorial and nΓ(n)

    Homework Statement Prove that for a positive integer, p: https://www.physicsforums.com/posts/5859454/I've tried this to little avail for the better part of an hour - I know there's a double factorial somewhere down the line but I've been unable to expand for the correct expression in terms of...
  15. Derek Hart

    B Proof that (x^n)/n has a limit of 0 at infinity

    I understand that the standard proof is a bit different from my own, but I want to know if my reasoning is valid. PROOF: Firstly, I assume that x is positive. I then consider p = inf{n∈ℕ : n>x} . In other words, I choose "p" to be the smallest natural number greater than x. If we choose n>p...
  16. Terrestrial officer

    B Factorial and the commutative property of multiplication

    In how many different ways can we arrange three letters A, B, and C? There are three candidates for the third position that leaves the two remaining letters for the second position and so 3 times 2 is 6 and One is the multiplicative identity I am astonished by The commutative property of...
  17. terryds

    What is the value of the harmonic factorial series sum?

    Homework Statement What is the value of ## \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + ... ## ? Homework Equations [/B] I have no idea since it's neither a geometric nor arithmatic seriesThe Attempt at a Solution [/B] My Calculus purcell book tells me that it is e - 1 ≈...
  18. B

    Compare & Determine: The 1/n! Series Convergence/Divergence

    Homework Statement Determine whether the series converges or diverges. ∞ ∑ 1/n! n=1 Homework Equations If ∑bn is convergent and an≤bn for all n, then ∑an is also convergent. Suppose that ∑an and ∑bn are series with positive terms. If lim an = C n→∞ bn where c is finite number and c>o...
  19. S

    Hexadecimal and factorial problem

    Homework Statement Hello all, I am trying to determine the last hexadecimal digit of a sum of rather large factorials. To start, I have the sum 990! + 991! +...+1000!. I am trying to find the last hex digit of a larger sum than this, but I think all I need is a push in the right direction...
  20. T

    MHB Evaluating a limit with a factorial

    We are starting sequences, and in one of the examples we have this limit: $$\lim_{{n}\to{\infty}} \frac{R^n}{n!}$$ We let $M$ equal a non-negative integer such that $ M \le R < M + 1$ I don't get the following step: For $n > M$, we write $Rn/n!$ as a product of n factors: $$\frac{R^n}{n!}...
  21. DuckAmuck

    I Identify Factorial: Is It Possible?

    Is there a way to identify a factorial without referring to computation of a factorial? For example, is there a way to identify 5040 as a factorial and a way to identify 5050 as not a factorial?
  22. H

    I Simplifying a fractional factorial with variables

    I am trying to work through a simplication of this factorial with variables: (n/2)!/[(n+2)/2]! I get, 2[n(n-1)]/2[(n+2)(n+1)n(n-1)] cancelling the 2[n(n-1)] leaves me with 1/[(n+2)(n+1)] However, Wolfram Alpha tells me this can be simplified as 2/(n+2) and I don't see that. Thanks
  23. gsmtiger18

    Limits of sequences involving factorial statements

    Homework Statement I have to determine whether or not the following sequence is convergent, and if it is convergent, I have to find the limit. an = (-2)n / (n!) In solving this problem, I am not allowed to use any form or variation of the Ratio Test. 2. The attempt at a solution I was...
  24. I

    MHB What is the last odd digit in the factorial sequence?

    Hi guys, my english is very bad but let me try translate the question: Let $ a_n=\frac{(n+9)!}{(n-1)!} $ . Let k the lesser natural number since that the first digit (on the right side) after all the zeros of $ a_k $ is odd. Example: $ a_k =$ 4230000000 or $ a_k =$ 62345000 This odd digit...
  25. karush

    MHB How to Strike Through Text in LaTeX?

    Simplify. Assume that $n$ and $m$ are positive integers, $a>b$, and $a>2$.$\frac{\left(a+1\right)!}{\left(a-2\right)!}$ was helping a friend with this but was clueless I know that n! $=n(n-1)(n-2) ... $
  26. D

    How to treat negative integer factorial function?

    Dear all, How to deal with negative integer factorial functions? I mean what expression formula can be substituted for this?
  27. Saracen Rue

    A googolplex expressed in factorial form?

    Does anybody know what the factorial form of a googolplex would be?
  28. A

    Factorial Q: How to Get (n+1)!

    How do you get (n + 1)! = (n + 1)(n)(n - 1)(n - 2) ... 3 ⋅ 2 ⋅ 1 ? Isn't (n + 1)! = (n + 1) ⋅ (n + 1 - 1) ⋅ (n + 1 - 2) ⋅ (n + 1 - 3) ⋅ (n + 1 - 4) ... and so on?
  29. D

    Factorization of floor functions of fractions

    hey so if you are taking a floor function of a fraction >1, is there any way to predict anything about it's factorization? what about when the numerator is a factorial and the denominator is made up of factors that divide said factorial but to larger exponents then those that divide the...
  30. Cosmos

    The Value of 100! to 110! Factorial Problem

    What do you think is the value of 100!-101!+102!-103!...-109!+110! :biggrin:
  31. ognik

    MHB Is the Double Factorial Series Convergent with Stirling's Asymptotic Formula?

    Hi, question is - show that the following series is convergent: $ \sum_{s}^{} \frac{(2s-1)!}{(2s)!(2s+1)}$ Hint: Stirlings asymptotic formula - which I find is : $n! = \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n $ I can see how this formula would simplify - but can't see how it relates to the...
  32. Odious Suspect

    Binomial Coefficient Factorial Derivation

    A few decades ago my algebra teacher showed how to construct the expression for binomial coefficients. If I start with Pascal's recursion, and propose C(n,k)=n!/k!(n-k)!, I can prove it to be so through induction. But that doesn't give me that happy feeling that comes with understanding. It...
  33. T

    How can I prove factorial equations involving difficult questions b and c?

    number 15 questions b and c are giving me a very hard time. I have tried expanding them then factoring out the common terms but somehow not getting it to be proven. detailed help will be appreciated.
  34. Md. Abde Mannaf

    Fortran Fortran 90 Program Factorial: Fixing Incorrect Output

    program factorial implicit none integer::fact,i,n print*,'enter the value of n' read*,n fact=1 do i=1,n fact=fact*i end do print*,'factorial is ',fact end program when input n largest number then answer is incorrect. how to solve
  35. C

    Convergence or Divergence of Factorial Series

    How can I find out if 1/n! is divergent or convergent? I cannot solve it using integral test because the expression contains a factorial. I also tried solving it using Divergence test. The limit of 1/n! as n approaches infinity is zero. So it follows that no information can be obtained using...
  36. T

    MHB Find the Value of n in a Unique Factorial Problem with (n+1)!/(n-1)! = 56

    Ok, I had never seen a factorial problem like this, and the answer(n=7) didn't help me much in understand the solution either. If (n+1)!/(n-1)! = 56 , what's the value of n?
  37. ironman

    Solving Limits: Find Interval & Radius of Convergence

    Homework Statement [/B] I have to find the radius of convergence and convergence interval. So for what x's the series converge. The answer is supposed to be for every real number. So the interval is: (-∞, ∞). So that must mean that the limit L = 0. So the radius of convergence [ which is...
  38. M

    MHB Correcting a 2 x 3 Factorial ANOVA Chart

    Have I completed this 2 x 3 factorial ANOVA chart correctly? A sample of ¬N = 36 is recruited to participate in a study about speech errors. The researchers believe that there is an interaction between whether the speaker is distracted (distracted or not distracted) and the difficulty of the...
  39. M

    MHB Program to compute the factorial of n

    Hey! :o I am asked to write a RAM (Random Access Machines) program to compute $n!$ given input $n$. Could you give me some hints how I could do that?? (Wondering)
  40. powerof

    Limit problem involving a double factorial

    Homework Statement Solve the following limit: $$ \lim_{n\rightarrow \infty }n\cdot\left ( \frac{2\cdot4\cdot6 \cdots (2n-2)}{1\cdot3\cdot5\cdots (2n-1)} \right )^{2}$$ The Attempt at a Solution I don't know where to begin. Until know I've encountered limits which I could deal with in some way...
  41. K

    MHB Interesting identity arising from fractional factorial design of resolution III

    I am learning about statistical design of experiments, and in the process of mathematically rigorizing the concepts behind fractional factorial designs of resolution III, I derived an interesting equation: $$k = \sum_{i=1}^{3}{\lceil{\log_2{k}}\rceil \choose i},$$ for which the solutions $k$...
  42. H

    The factorial of a rational number, the gamma function not used

    My first question is: is this formula (at the bottom) a known formula? In this subject i haven't explained how i build up the formula. So far i think it is equal to the gamma function of Euler with \Gamma\left(\frac{m_1}{m_2}+1\right)= \frac{m_1}{m_2}\ ! with m_1 , m_2 \in...
  43. M

    Relating integral of powers of Sin b/w 0 and pi/2 to factorial form

    Our integral \int\limits_0^{\pi/2} \sin^{2a+1}(x)\,dx Has a Factorial Form: {(2^a a!)}^2 \over (2a+1)! What is the process behind going from that integral to that factorial form? My approach which is not very insightful: I used mathematica to calculate the integral to return...
  44. A

    MHB How to calculate complicated factorial

    Hello. I know what factorial means but how do I calculate this? Could someone explain to me on how to do it?
  45. Runei

    Integral definition of factorial

    I'm watching V. Balakrishnan's video lectures on Classical Physics, and right now he's going through statistical mechanics. In that regards he's talking about Stirlings formula, and at one point, he wrote an integral definition of the factorial like the following n! =...
  46. vanceEE

    Simplifying Factorials: How to Simplify the Expression (2n-1)! / (2n+1)!

    Homework Statement $$ \frac{(2n-1)!}{(2n+1)!} $$ How do you simplify this factorial?
  47. A

    How to Calculate the Factorial of Avogadro Number?

    can anyone explain to me how to find the factorial of avogadro number ? what is its value ?
  48. A

    Why is the Factorial of Zero Defined as One?

    can u give me the method to find the factorial of zero ?
  49. I

    MHB Prove that factorial n is less than or equal to n raised to n

    HelloI wish to prove that \[ \forall\;n\in \mathbb{N}\; n! \leqslant n^n \] First we let \(n\) be arbitrary. Now I first write \( n! \) as \( n\cdot(n-1)\cdot(n-2)\cdots 3\cdot 2\cdot 1\). Now we see that \[ n \geqslant (n-1)\;; n \geqslant (n-2)\;\ldots ;n \geqslant n- (n-1) \] So we get \[...