What is Factorial: Definition and 161 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. J

    MHB Total ordered pair in factorial equation

    (A) Total no. of integer ordered pairs $(x,y)$ in $x^2-y! = 2001$ (B) Total no. of integer ordered pairs $(x,y)$ in $x^2-y! = 2013$ My Trail :: (A) Given $x^2-y! = 2001 = 3 \times 23 \times 29 \Rightarrow x^2 -y! = \left(3\times 23 \times 29\right)$ means $(x^2-y!)$ must be divisible by...
  2. anemone

    MHB Sum of Factorial Series: Find the Answer!

    Find the exact value of the series \frac{1}{4!}+\frac{4!}{8!}+\frac{8!}{12!}+\frac{12!}{16!}+\cdots\cdots
  3. Seydlitz

    Convergence of Factorial Function: Is Stirling Approximation Necessary?

    Homework Statement $$\lim_{n \to \infty}\frac{10^n}{n!}$$Homework Equations Do I need to use something as advanced as Stirling approximation? This question appears in page 5 of Boas Mathematical Methods, there's no prior discussion on factorial or convergence test and etc. The Attempt at a...
  4. J

    Is this Factorial Sequence Increasing, Decreasing, or Neither?

    Homework Statement These trickly little buggers always seem to confuse me. I need to find out whether or not the sequence is increasing, decreasing or neither. An=(n!)2/(2n)! Homework Equations The Attempt at a Solution I'm pretty sure that it's a decreasing sequences but when I expand and...
  5. B

    Factorial Notation for Gamma Function Expression with Fractional Input?

    Hi members, gamma (n+1)=n! Now I have the following: gamma(2(n+3/2))=gamma(2n+1) What is factorial notation for this?? Thank you
  6. F

    Using a factorial moment generating function to find probability func.

    Homework Statement Hi everyone! Me and my colleague are working our way through Harold J Larson's "Introduction to Probability Theory and Statistical Inference: Third Edition", and we found something interesting. We both have the same edition of the text, but mine is slightly newer?, and...
  7. U

    Comp Sci To write a C++ program to find the factorial of a number

    Homework Statement To write a C++ program to find the factorial of the number input 'a'. Homework Equations the factorial evaluated is denoted as 'f' i is the counter The Attempt at a Solution Pls check if my program is correct...(here f is the value of factorial and a is the...
  8. P

    Hypothesis Testing: Factorial ANOVA

    Hello! So I had to propose a research design as an assignment and I chose to look at cognitive function in an exercise group vs. a control group over the period of 10 years. -subjects are tested at the beginning and end of each year The groups were not randomized on perfectly (I used two...
  9. M

    Fascinating Factorials: Exploring the Mystery of the Last x! in (x + y)!

    Why is it that (x + y)!=(x + y)(x + y - 1)(x + y - 2)...(x + 1)x! Where did the last "x!" come from? Thanks
  10. Z

    Convergence of Factorial Functions at Infinity

    Homework Statement Show that the folowing holds: \lim_{n\to\infty} \frac{(n!)^2 4^n}{(2n)!} = \inftyHomework Equations It can be shown that \frac{(n!)^2 4^n}{(2n)!} = \prod_{k=1}^n \frac{4k}{k+n}The Attempt at a Solution If I can proove that \lim_{n\to\infty} \ln\left[\frac{(n!)^2...
  11. ShayanJ

    Is infinity factorial equal to the square root of 2 pi?

    I was studying about infinite products that I got to the relation below in http://mathworld.wolfram.com/InfiniteProduct.html \infty != \sqrt{2 \pi} It really surprised me so I tried to find a proof but couldn't. I tried to take the limit of n! but it was infinity.Also the limit of...
  12. J

    Difficult z transform with a factorial

    Homework Statement find z transform of: x[n] = (1 / n!) *u[n] u[n] is the unit step Homework Equations z transform equationX(z) = Ʃ x[n] * z-n summation is from -∞ to +∞ The Attempt at a Solutioncancel the u[n] by changing the bounds of the summation now it is from 0 to +∞It's at this...
  13. P

    How can I simplify this expression involving summation and factorials?

    I need to simplify this expression and I don't know how to deal with the factorials in the sum e^{-(\lambda + \mu)}\sum_{k=0}^w \frac{\lambda^k \mu^{(w-k)}}{k!(w-k)!} Can anybody give me a hint on how to sum over the factorials?
  14. iVenky

    Is gamma function derivative of factorial?

    I was searching for derivative of factorial. Many say that gamma function is the derivative of the factorial. Is that true because I searched about gamma function and it doesn't say anything like that. Thanks a lot
  15. B

    Writing The Limit Of A Sequence With A Factorial

    Homework Statement The sequence is a_n = \frac{n^P}{e^n} Homework Equations The Attempt at a Solution If I did L'Hopital's rule P times, would the final product look like: P!\times lim_{n \rightarrow \infty} = \frac{1}{e^n} ?
  16. S

    Factorial of a negative number

    Is it possible to find the factorial of a negative real number ? If possible, then what is its practical application ?
  17. B

    Infinite Sequence Involving A Factorial

    Homework Statement a_n = \frac{(2n -1)!}{(2n)^n} Homework Equations The Attempt at a Solution I am not exactly sure how to solve this problem.
  18. B

    Convergence of a factorial function

    The sum of "(n+3)!/(3n+2)!" with n=1 to n=inf. How do I find if it converges or diverges by using one of the tests(ratio, roots series, divergence, etc)?
  19. P

    Factoring with factorial exponents.

    Homework Statement Factor x + 5+ 6x^-1 Factor x^(3/2) + 2x^(1/2) - 8x^(-1/2) Homework Equations None given. The Attempt at a Solution I have tried factoring normally, it's just not working out though (for either part of the question.) I've never had to deal with this in a math...
  20. D

    Probability - factorial algebraic manipulation

    I saw this in my book and I'm having a difficult time figuring out how this formula was manipulated algebraically to equal the other side. I know that it works because I've used it several times but I can't show that it is true. \frac{n(n-1)...(n-r+1)}{r!} = \frac {n!}{(n-r)!r!} One of...
  21. P

    Factorial identity question

    \frac{2 i}{(2 i + 1)!} = \frac{1}{(2 i)!} - \frac{1}{(2 i + 1)!} Could anybody please show what it is that needs to be done on LHS to get to RHS in this identity.
  22. FeDeX_LaTeX

    Positive Int Solns for Factorial Equation: a!b!=a!+b!+c!

    Homework Statement Find all positive integer solutions for a, b and c to the equation: a!b! = a! + b! + c!Homework Equations n! = n(n-1)(n-2)... The Attempt at a Solution I'm not having much progress with this. I've tried rewriting it as (a! - 1)(b! - 1) = c! + 1 and I've found one...
  23. A

    What's relation between factorial and tensor components.?

    maybe its simple permutation and combination technique... I ve tried to resolve it on my own but I couldn't...The textbook which I read simply mentions it to be obvious..can anyone please elucidate me the logic behind factorials of "independent components of tensor curl" and levi cita Can...
  24. D

    State hypothesis for 2k factorial design

    How to state null and alternative hypothesis for 2k factorial design of two factors? The Attempt at a Solution All i can think of right now is null hypothesis - Factor A effects not significant. alternative hypothesis - Factor A effects significant null hypothesis - Factor B effects...
  25. S

    Proof that series diverges ? (Defn of factorial ?)

    Ʃ \frac{(4n)!}{n!(2n)!} I tried the ratio test, but if my solution has a limit which goes to infinity. Hence the test is inconclusive. (Or does L->inf imply divergence as L>1 ?) Any ideas what other test might be useful ? PS: I have used the idea that (2(n+1))! = (2n+2)! = (2n+2)(2n+1)(2n)...
  26. A

    Simple factorial and matrix question

    if I have n slots [1,2,3,4,5,...n] where I must use numbers from 1 through n, and I have to ensure that none of them were reused... the way I figure to make sure that none of the numbers are reused is to multiply the values in each slot and compare them against n! e.g. [1,2,3,4,5] would be...
  27. C

    Can Inequality with Factorial Be Proven without Induction?

    Homework Statement \frac{1^2*3^2*5^2...(2n-1)^2}{2^2*4^2*6^2...(2n)^2}<\frac{1}{2n+1} Edit: Must be proven without using induction. Homework Equations The Attempt at a Solution I understand the LHS is the same thing as \frac{(2n-1)!}{(2n)!} And (2n)! = k!2^k & (2n-1)! =...
  28. P

    Factorial coefficient simplification

    Hey, I've been trying to find a way to simplify these two series coefficients, \begin{align} & {{a}_{4n}}={{a}_{o}}\frac{2!6!...(n-6)!(n-2)}{4!8!...(n-4)!n!} \\ & {{a}_{4n+1}}={{a}_{1}}\frac{3!7!...(n-6)!(n-2)}{5!9!...(n-4)!n!} \\ & E.g. \\ &...
  29. S

    Convergence/Divergence of a Geometric series with a Factorial

    Homework Statement Determine if the sequence {an} below converges or diverges. Find the limit of each convergent sequence an = n!/nn Hint: Compare with 1/n . Find the limit of the sequence {an} if it converges. I missed the lesson on factorials, and the book is useless. Sorry...
  30. A

    MATLAB How to write factorial in matlab

    Dear Fellows, I want to write rational numbers factorial in my MATLAB program for integers factorial command is factorial(n) but for rational numbers such as 0.1 or 0.2 this command is not working can anyone guide me to solve this issue ? Regards
  31. T

    How to solve factorial related problems

    Hi all, I know what factorials are, obviously, and permutations and combinations, but what I Don't know is, given a problem with factorials in it, is there a general format for solving? For instance, an old AMC 10 problem: 10B-#11. What is the tens digit in the sum 7! + 8! + 9! + . . . +...
  32. D

    Help Factorial Partial Fraction Decomposition

    Homework Statement Show that n/(n+1)!=(1/n)-(1/(n+1)!) I am totally lost on the algebraic steps taken to come to this conclusion. It is for an Infinite series. Thanks
  33. E

    Factorial question in a power series solution

    Hello, I've been working on solving the equation y''-2xy'+2py=0. where p is a positive integer. I've assumed y=\sum a_{n}x^{n} for n=0 to inf I'm getting two formulas for a_{n} One is for odd n, the other for even n, related to a_{0} and a_{1} However, the relation involves something that...
  34. N

    Factorial simplification problem

    Homework Statement In my calculus book they have the following simplification: \frac{k!}{(k+1)!} = \frac{1}{k+1} I do not quite understand how that's occured.
  35. M

    Proving Double Factorial: (2n)!=2^nn!

    How to prove (2n)!=2^nn! for example (2)!=2^11!=2 (4)!=2^22!=8 ... (2n)!=2n(2n-2)(2n-4)...=2^3n(n-1)(n-2)... I see that by intiution but I don't know how to write prove.
  36. J

    Expressing double factorial for odd integers

    Homework Statement Express \frac{1}{(2n+1)!} as the following \frac{(-2)^{n}n!x^{2n+1}}{(2n+1)!} where 0 <= n <= infinity Homework Equations The double factorial for odd integers is (2n+1)! = (2n+1)(2n-1)(2n-3)...1 where 0 <= n <= infinity The Attempt at a Solution...
  37. E

    Comp Sci C++ Factorial Calculator Exercise Solution

    Homework Statement Write a C++ code factorial.cc that calculates the factorial of an integer by direct multiplication. Start with your code in exercise (3), and modify it so that it gets "n" from the user and prints out n! = 1*2*3*...*(n-1)*n. To do this, use a "for" loop: int nfac...
  38. D

    Summing Factorials: Solving the Homework Statement

    Homework Statement \sum_{n=0}^{100} 1/n!(100-n)! The Attempt at a Solution Other then obvious attempts to make sense of the equation's incremental and decrements divisor, I can't figure out where to start with this question. Some assistance would be greatly appreciated.
  39. F

    Infinite limit of factorial series

    Homework Statement The Attempt at a Solution I have no idea where to start from. I know that e = 1/0! + 1/1! + 1/2! + 1/3! + ... + 1/n! + ... and tried to separate the factorial off the geometric series but had no success.
  40. D

    What is the Limit of (1/n!)^(1/3n) as n Approaches Infinity?

    Homework Statement I need to evaluate the following limit: lim_{n\rightarrow\infty}\left(\frac{1}{n!}\right)^{\frac{1}{3n}} Now, I think I have solved it, but my methods are not rigorous. I am looking to see if my assumptions are sound and if there is an easier way to solve this limit...
  41. T

    Shifted Factorial Homework: Showing \sum_{n=0}^{\infty}

    Homework Statement I've got to show \sum_{n=0}^{\infty} \frac{(a)_n(-1)_n}{(c)_n n!} = \frac{c-a}{c} where \displaystyle (a)_n = \frac{\Gamma(a+n)}{\Gamma(a)} = a(a+1)...(a+n-1) is the shifted factorial (Pochhammer symbol). The Attempt at a Solution I've been informed that (-1)_n =...
  42. R

    Therefore, (mn)! is divisible by m!(n!)^m.

    Hi, we know that for all interger m, n_1!\cdots n_k! divides m! where n_1+\cdots+n_k=m. Now I want to show that m! (n!)^m divides (mn)!. We see that (n!)^m divides (mn)! since \overbrace{n+\cdots+n}^{m-terms}=mn Also m!(nm-m)! divides (mn)! similarly. But how could I show my required...
  43. M

    Proving convergence of factorial w/o Ratio Test

    Homework Statement Determine whether 1/n! diverges or converges. So far, we have only learned the comparison tests, p-series, geometric series, divergence test, and integral test, so I can only use these tests to prove it. Homework Equations N/a The Attempt at a Solution I...
  44. D

    Are these factorial statements accurate?

    so, 6! = 6*5*4*3*2*1 (n+2)! = (n+2)(n+1)(n)! (2n+2)! = (2n+2)(2n+1)(2n)! (500n+3)! = (500n+3)(500n+2)(500n+1)(500n)! Are all these statements correct?
  45. P

    Calculating (n+2)! quickly and accurately - Help Needed!

    I know that n!=(1)(2)(3)...(n-1)(n) I am confused how I can figure out what (n+2)! is in order to divide it by n! How can I figure (n+2)! out and what is it, so that I won't have to ask if what I got is right. Thanks!
  46. A

    Can You Find a Finite Expression for the Summation of Factorial Terms?

    Hi, I've been trying to make another expression for, m \sum(k!) = f(m) k=0 I did one expression that is, f(m) = 2(3(4(5...((m-1)(m+1)+1)...+1)+1)+1)+1 For instance, f(4) = 2(3(4+1)+1)+1 Can you hint me on finding a finite expression for the above expression? Thanks for help.
  47. K

    Expand Factorials: (n-2)! & (n-n')!

    I know how to expand a factorial of the form (n-2)!. How would one expand (n-n')! ? Would it just be (n-n')? thanks in advance
  48. S

    Solving an Unknown Factorial

    Is there a method to solve an unknown factorial algebraically? Homework Statement 5040 = a! I found the solution a = 7 through trial and error, was just wondering if a more elegant method exists.
  49. Z

    How many ways to rearrange distinct objects (factorial)

    Homework Statement Hello. This is the gist of the question: I have 0 < k < n and I need a formula to show the number of sets of exactly k integers, each chosen from 1, ... , n. Homework Equations n! is the number of distinct ways to rearrange n objects into n slots. The...
  50. 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.
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