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. L

    Solve N Factorial Puzzle with 50 Zeros

    Homework Statement N is divisible by 4. N! has exactly 50 zeros. Find N. Homework Equations In case anyone younger doesn't know, Y!=Yx(Y-1)x(Y-2)x(Y-3)x...x3x2x1 The Attempt at a Solution No idea.
  2. K

    Prime Factorial Proof: Existence of a Prime Between n and n!

    Prove or disprove: If n is an integer and n > 2, then there exists a prime p such that n < p < n!.
  3. M

    Factorial : n/(n-k) = n(n-1)(n-2) (n-k+1) - why?

    Why is the equation (A) n!/(n-k)! = n(n-1)(n-2)...(n-k+1) true? For example, let n=4 and k=2, then 4!/2! = 4x3x2x1 / 2x1 = 4x3 = 12. I understand this example, but I can't make the connection with this and the right-hand-side of equation (A). For example, why is our...
  4. E

    Explaining Ultimate Factorial Value in x^n Integer Series

    In the situation where differences between consecutive squares, (or consecutive cubes, consecutive x^4, etc.) are calculated, then the differences between those differences are calculated, and then the differences of those differences, and so on until you reach a constant number at a deep...
  5. E

    Number of zeros in factorial of a number

    Homework Statement This is not a homework problem. I was just wondering the logic and intuition behind the "formula" for finding the number of zeros given the factorial of a number. Homework Equations Formula : round(n/5) + round(n/25) + round(n/125)+...+round(n/5^n) Here, round...
  6. J

    Understanding the Proof: Factorial of 0 is 1

    Factorial of 0=1 ?! Can anyone please explain to me the proof that the factorial of 0 is 1?
  7. E

    (n+1)=(n+1)n factorial problem

    First of all apologize for my english, I'm french and I'll do my best to be understandable. So my question is about factorials. how do you manage to say that (n+1)!=(n+1)n! ? I tried to develop this but my brain is just not able to understand how I'm suppose to do. Could someone please show...
  8. M

    Ratio Test inconclusive with factorial

    Homework Statement \sum(n-1)!/(n+2)! 2. The attempt at a solution I tried the ratio test and came up with the lim_{}n\rightarrow\infty n/(n+3) = 1 which gives no information on convergence or divergence. I'm trying to find absolute or conditional convergence so what else can I do?
  9. N

    Exploring Factorial Properties: Representations of n!, (n+1)!, and (n-1)!

    Id like to know some basic representations of factorials n!, (n+1)!,(n-1)! ext..
  10. D

    Classifying an Alternating Factorial Series

    Homework Statement Classify the series as absolute convergent, conditionally convergent, or divergent. \Sigma^{\infty}_{k = 1} (-1)^{k-1}\frac{k!}{(2k-1)!} Homework Equations The Alternating Series Test: conditions for convergence decreasing lim --> infinity ak = 0 The...
  11. M

    N factorial - general question

    Hello everyone! Homework Statement n!/(n-r)! = n(n-1)(n-2)...(n-r+1) where r is the number of objects we want from n distinct objects (3 billiard balls out of 16) I don't understand what the last term in the expansion means, the (n-r+1). For example, suppose we have 5 distinct...
  12. T

    Proving Lim (2^n)/sqrt(n!) = 0

    So I've been asked to prove that: lim (n-->infinity) [2^n]/sqrt(n!) = 0 I've tried fiddling with Stirling and L'Hospital, but can't find my way through it. Any thoughts?
  13. C

    Finding the Derivative of Factorial Function

    Hi, I want to find the derivative of factorial function f(x)=x! and i found this integral, f(x)=x!=\int_{0}^{\infty}e^{-t}t^xdt when i take derivative of this \frac{d}{dx}f(x)=\frac{d}{dx}\int_{0}^{\infty}e^{-t}t^xdt=\int_{0}^{\infty}e^{-t}t^xlnxdx How do i find this integral? Please...
  14. E

    Determining convergence of infinite series with factorial without ratio test

    Homework Statement Determine whether the series below is convergent or not: \sum 7*\frac{n!}{n^{n-10}} n=8 and the series goes to infinity (Sorry, I couldn't get the formatting correct.) Homework Equations n/aThe Attempt at a Solution Well, originally I thought the series was divergent...
  15. W

    Extending addictive factorial?

    If we define an addictive factorial for any integer n: f(n) = n + (n-1) + (n-2) ... 0 1!+ = 1 2!+ = 2+1 = 3 3!+ = 3+2+1 = 6 4!+ = 4+3+2+1 = 10 5!+ = 15 is it possible to extend it to real or possibly complex numbers by analytic continuation? just like the gamma function extends the factorial.
  16. A

    Try using a sterling approximation on the factorial

    Can somebody demonstrate : \[ \frac{n}{{\sqrt[n]{{n!}}}} < \left( {1 + \frac{1}{n}} \right)^n \] ITS not A HOMEWORK
  17. N

    Simplifying the Factorial: How is it Done?

    Can somebody please explain to me this simplification and how it's done? \frac{n!}{(2n)!} = \frac{1}{(2n)(2n-1)...(n+1)} Thanks a lot.
  18. G

    Determine Limit of Factorial Sequence a_n

    Homework Statement Determine the divergence or the convergence of the sequence. If it converges find its limit. a_{n} = (\frac{(n)!}{2n!+1}) The Attempt at a Solution All I know about factorials is for example 4! = 1*2*3*4. So as far as limits go I'm clueless. please help!
  19. K

    Solving for Time in Scalar Product Equation

    Homework Statement I am solving for t using this equation: d = !(v + 0)t; My d=.984 and v=1.9 I don't know what to do with the "!" Homework Equations d = !(v + 0)t; The Attempt at a Solution In a problem with different numbers I was able to solve for t by dividing and...
  20. Saladsamurai

    C/C++ C++ Programmer Defined Factorial Function

    Hi I am relatively new to C++ and I am having a little trouble understanding, in detail, the logic of this recursive function. Can someone tell me if my reasoning this out is correct? int fact (int n) { if (n==1) return 1; else return (n*fact(n-1)); } So if the...
  21. S

    Help Solving a Factorial Program with Infinite FOR Loop

    i m unable 2 get the output for the followin c program. The factorial program is correct i think there is problem in the FOR loop..guess its goin infinite...pls help #include<stdio.h> #include<math.h> factorial(int); main() { int n,count,i; float number,x,series; printf("nter number...
  22. D

    Solving Factorial Algebra: P(n, 4) = 40[P(n-1, 2)]

    Homework Statement P(n, 4) = 40[P(n-1, 2)] Homework Equations ? The Attempt at a Solution I boiled this down to the equation n!/(n-4)! = 40[(n-1)!/(n-3)!]. The problem is, I have no idea how to perform the correct operations on these factorials. I found the answer to be n = 8...
  23. F

    How can I simplify this factorial expression?

    Homework Statement \frac{(kn)!}{(kn+k)!} I was thinking: (kn)! = 1*2*3...(kn) (kn+k)! = 1*2*3...(kn)(kn+k) and I would be left with 1/kn+k But my book has the answer as: \frac{1}{(kn+k)(kn+k-1)...(kn+1)} How can I arrive to that?
  24. K

    How to Simplify Factorials with a Proof for k*(k!)=(k+1)!-1?

    Homework Statement I'm trying to prove k*(k!)=(k+1)!-1 Homework Equations The Attempt at a Solution This is how far I've gotten: k[k(k-1)(k-2)...1)]
  25. T

    Mathematica Comparing Gamma & Factorial Functions in Mathematica

    Why does Mathematica sometimes give Gamma, the funtion and sometimes !, the factorial? What is the differences?
  26. K

    (4-digit number)*(6-digit number) equals a factorial

    Substitute each of the letters by a different decimal digit from 0 to 9 to satisfy this cryptarithmetic equation: (ABCD)*(EFEGBH) = (EC)! Note: None of A and E can be zero.
  27. U

    Understanding the First Part of an Inequality Factorial

    Why is the first part of this inequality true? 1/(n+1)! [ (1 +1/(n+1) +1/(n+1)^{2} +...+ 1/(n+1)^{k} ] < 1/(n!n) < 1/n
  28. N

    Lim n to infinity for factorial

    Homework Statement lim n -> infinity for (n!)^(1/n) Homework Equations The Attempt at a Solution hmm, i know that lim n approaches infinity, (n)^(1/n) will go to 1, but issit the same for n!?
  29. rocomath

    Self studying, similar to a Factorial?

    I don't have much experience in this, and hopefully someone can recommend the right type of book that I need to look thru in order to solve future problems like this. I have to have a formula for the nth derivative. I have like in the numerator, but Idk how to express it properly. 5 5 x 10...
  30. J

    How Do You Calculate the Limit of a Factorial?

    Please can anybody help me find this limit?
  31. P

    Factorial Function: Does it Hold for Non-Integer N?

    Homework Statement I was just wondering if n!=n(n-1)! is completely general. Does it hold even for non-integer n? Homework Equations The Attempt at a Solution
  32. M

    Is My Understanding of Factorial (2n+1)! Correct?

    For the factorial (2n+1)!, I thought the previous term is going to be (2(n-1)+1), which is equal to (2n-1). Thus (2n+1)!= (2n+1)(2n-1)! However, in the textbook, they have it as . a_n= \frac{(2n-1)!}{(2n+1)!}=\frac{(2n-1)!}{(2n+1)(2n)(2n-1)!} Are they wrong or I am wrong? Thanks!
  33. R

    Simplifying Summation and Factorial

    I was looking at the web page containing a derivation for the Poisson distribution: http://en.wikipedia.org/wiki/Poisson_distribution which derives it as the limiting case of the binomial distribution. There is a simplification step which I am missing, which is the step(s) between...
  34. I

    Can the Squeeze Theorem Determine the Limit of n^n/n! as x Approaches 0?

    [SOLVED] Factorial Limits Homework Statement lim n^n x->00 n! Homework Equations Instructor said to use the Squeeze theorem. The Attempt at a Solution So far I have not been able to come up with much. I have looked at breaking the top apart into (n)(n)(n)...(n) and the bottom into...
  35. D

    Convergence of a series with factorial

    Homework Statement My task is to determine whether the following series is convergent or not. \sum_{n=1}^{\infty}\left(\frac{n}{e}\right)^{n}\frac{1}{n!} The Attempt at a Solution The limit of the terms is zero, so the series might be convergent. I tried the ratio test, but the...
  36. B

    How can (eqn.1) be simplified to (eqn.2) using factorials and summation?

    Hello all! In solving some math problems, I encountered the following sum: \sum_{k=1}^{r+1} kb \frac{r!}{(r-k+1)!} \frac{(b+r-k)!}{(b+r)!}. \quad \mbox{(eqn.1)} Now, I have asked Maple to calculate the above sum for me, and the answer takes a very simple form: \frac{b+r+1}{b+1}. \quad...
  37. S

    What is the limit of the sequence (Xn) = (n!)^(1/n)?

    I'm working on the limit of the sequence (Xn) = (n!)^(1/n) Pretty sure it diverges as n goes to infinity, but unsure how to show it. Any hint would ge greatly appreciated.
  38. gabee

    Summation analogue of the factorial operation?

    Is there such a thing? The factorial is usually defined as n! = \prod_{k=1}^n k if k is a natural number greater than or equal to 1. Is there an operation that is defined as \sum_{k=0}^n k if one wants to find, for instance, something like 5+4+3+2+1? I ask because I was thinking about...
  39. K

    Limit with Factorial: Understanding 3/(n+1) Limit

    Homework Statement Why does the limit as n -> infinity of [3^(n+1)]/(n+1)!] * n!/(3^n) equal the limit as n -> infinity of 3/(n+1)? Homework Equations The Attempt at a Solution I have never encountered this before.
  40. P

    Reducing Factorials: Is the Answer 1/[2n*2n+1]?

    Homework Statement (2n-1)!/(2n+1)! Homework Equations The Attempt at a Solution ...2n-2*2n-1 ------------ (pretend that's a divider) ...2n-2*2n-1*2n*2n+1 Is the answer 1/[2n*2n+1]?
  41. T

    Factorial Question | Solve k(n-1)! Equation

    Hi, I have a question about factorials that I'm hoping someone can help me with. I know that the factorial n! means the product of the integers from 1 to n, for example if I have 4! then this is equal to 4 x 3 x 2 x 1 = 24, but I have an equation which contains the term: k(n-1)! I am...
  42. A

    Is there a quicker way to calculate factorials?

    Hello. Do anyone know the shortcut or trick of taking factorial directly? for example : 9! = 9*8*7*6*5*4*3*2*1. it is very time consuming method to multiply all these terms to get answer. please tell me easy mathod. thank you
  43. U

    Is Zero Not Equal to One and Why?

    Just wondering: Why does 0!=1 ? Thanks
  44. T

    Solving CuTee's Factorial Problem with QValidator.h

    #include<qvalidator.h> //int input; void Factor::init(){ InputEdit->setValidator(new QDoubleValidator(InputEdit)); FactorNumber(); InputEdit->selectAll(); } int Factor::factorials(int n){ if(n<=1) return 1...
  45. F

    Solving Factorial Notation Questions: An Explanation

    Hey, I was wondering if someone could help me with a specific type of question that I can't seem to understand without an answer key. Anyway, it's rewriting expressions with factorial notation so that they no longer have factorial symbols. Example: Simplify without using the factorial...
  46. D

    Sum of Factorial Series with Sine Function: Understanding the Conversion

    I don't understand this conversion! \sum_{n=1}^\infty \frac{sin(n\pi /2)}{n!} = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} I know that the numerator of the left side is 0 when n is an even number. When n is odd, the numerator is either +1 or -1. But how do i continue?
  47. L

    How to prove the half number factorial formula?

    Any ideas on how to prove this? (n+\tfrac{1}{2})! = \sqrt{\pi} \prod_{k=0}^{n}\frac{2k+1}{2}
  48. H

    52 Factorial: How Many Zeros at the End?

    how many consecutive zeros are at the end of 52! ?
  49. Q

    Why is zero factorial equal to 1?

    Why is zero factorial equal to 1?
  50. murshid_islam

    Zeroes at the end of a FACTORIAL

    how can i find the number of zeroes at the end of 100! how can i find the number of zeroes at the end of n! thanks in advance.
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