What is Factorials: Definition and 90 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. W

    Simplifying formulas with factorials (2n-1)

    How would you simplify (2n+1)! Is the same as (2n-1)! ?
  2. T

    Understanding Factorials: A Combinatorics Primer

    In my combinatorics book, it's discussing inclusion-exclusion, and it says that n!-(n-1)! = (n-1)!*(n-1)! Can someone help me understand the rules of factorials? Thanks!
  3. T

    Factorials and Squares: Finding Solutions to a Unique Equation

    Homework Statement For some natural N the no of positive integers x satisfying the equation 1! + 2! + 3! + ... + x! = (N)2 is: A)one B)two C)infinite D)none The Attempt at a Solution I have no idea of how to start.. never came across such problems. By trial i have got two values...
  4. N

    Probability and Factorials Question

    Hi all. Does anybody know their stuff when it comes to math? More specifically calculating probability... i have a question about factorials and when they should be used. If you have let's say 2 events, the chance of event A is 1/2 the chance of event B is 1/3 you're just supposed to...
  5. N

    Prime Factorial Conjecture: Investigating p! Mod p2 for Prime Numbers

    Is there a name and/or proof for the following conjecture? "For any prime p, p! is congruent to p2-p modulo p2." Thanks much.
  6. P

    Understanding Factorials and Series Convergence in Mathematics

    I am unsure as to how factorials should be expanded. I have \sum\stackrel{1}{(2n!)} (if what was just typed did not make sense due to html error on my part, it is supposed to say the sum of 1/(2n)!) from n=1 to infinity. I did the ratio test and found the limit to be 0, which is less than...
  7. V

    Simplifying Factorials: Rules and Examples

    I need to figure out the following factorial \frac{297!}{98! * 199!} then take the logarithim of that Is there a rule that I can use to simplify the equation and get the same result? , I did another example where I used \frac{310!}{2!*299!} and I figured it out to be...
  8. P

    Evaluate Limit with Factorials: \lim_{n \to \infty}

    Anyone know of any method to evaluate this limit, \lim_{n \to \infty} \frac{n!}{\left(\frac{n+p}{2}\right)!\left(\frac{n-p}{2}\right)!}2^{-n} it seems to go to zero, but I have no way to be sure.
  9. J

    Factorial Series Convergence: Investigating the Sum of n!/10^n

    Homework Statement Show that \sum \frac{n!}{10^n} converges or diverges.(Note, I was unsure of how to format this via latex, so the summation is from n = 1 to infinity.)Homework Equations The root test: |\frac{a_n_+_1}{a_n}| The Attempt at a Solution a_n=\frac{n!}{10^n}...
  10. U

    An example of a series involving factorials

    The problem at hand: \inline{\sum_{k=1}^n \frac{(k+1)!}{(k+3)!}} Hence, find the limiting sum of the series, as n ---> infinity. Start this summation by expanding out the factorial to have a common factor of k!(k+1) as follows: \sum_{k=1}^n \frac{(k+1)!}{(k+3)!} = \sum_{k=1}^n...
  11. D

    Factorial Limits: Find the Limit of a Sequence as n Approaches Infinity

    Homework Statement Find the lim as n-->inf of the sequence {an}= 1x3x5x...x(2n-1) _______________ n! Homework Equations The Attempt at a Solution I rewrote it as ...(2n-3)(2n-2)(2n-1) __________________ n(n-1)(n-2)...2x1 which leads me to believe that is...
  12. I

    Reducing logarithms of factorials

    Homework Statement Part of a much bigger problem, but I am hung up on solving the following: ln\left [ \left(\frac{N+n}{2}\right ) ! \right ] = \left ( \frac{N+n+1}{2}\right) \frac{ln(N+n)}{2}\right ) I am trying to follow a proof in...
  13. E

    Statement about factorials that I don't understand

    I was reading and came across this statement: If t > 2n^2 is an integer, then t! > (n^2)^(t-n^2) I'm not sure why it is true. I don't know what equations are relevant. My feeling is that you don't need anything more than algebra, but perhaps it would also follow from the gamma function...
  14. Q

    Factorial Ratio for Interval of Convergence Calculation

    Homework Statement Hi! I need to find ratio (2n+1)!/(2n+3)! for interval of convergence calculation. Homework Equations 5! = 1*2*3*4*5The Attempt at a Solution i have no idea where to start since i have never dealt with factorials before.. if you just show me some kind off factorial...
  15. P

    Finding sum of Power Series with Factorials

    Homework Statement I am confused about how to find a sum of a power series, especially when it contains factorials and I can't quite get it to look like a geometric series. Is it the same thing as finding a limit (and then I would follow the various tests for convergence of the different...
  16. T

    Converge or diverge of factorials

    Homework Statement Determine whether the series converges or diverges. \sum\frac{37}{n!} I will just forget about the 37, and think of it as \sum\frac{1}{n!} I can try to decompose the n! n! = n(n-1)! n! = n ( n-1) (n-2)! n! = n(n-1)(n-2)(n-3)...2*1 so \sum\frac{1}{n!}...
  17. R

    Can the Definite Integral of (ln x)^n be Expressed Using Factorials?

    Homework Statement Show that the definite integral from 0 to 1 (ln x)^n dx = n!(-1)^n Homework Equations The Attempt at a Solution i tried to integrate by parts and kept going on and on but i don't know how to incorporate the factorial in the answer ...
  18. R

    Infinite sequence w/ factorials

    Hi, I was just looking at this problem with sequences and I was having a question about factorials. I understand that the factorials need to get smaller. I was just wondering what is the (3)(2)(1) in the problem symbolizing? Just that it will eventually reach the end? and what numbers...
  19. V

    Can you explain how to reduce factorials for the ratio test?

    Homework Statement This isn't a specific problem rather I don't know how to reduce factorials and this is giving me a hard time when I try the ratio test. For an example I'll use (2n+1)!/(2n+3)! Homework Equations The Attempt at a Solution I attempt it by writing out some...
  20. Alesak

    Can the equation x! + (x-3)! = 16x - 24 be solved for x?

    hello everyone, I usually post my questions on one small czech mathematical forum, but here is an equation no one knows how to "solve". I`ve came to it by accident, when I made an mistake in one combinatorics equation. x! + (x-3)! = 16x - 24 its fairly simple to solve in one way(x has...
  21. K

    Factorials approximation problem

    Homework Statement How is, [(N+Q)!Q!]/[(Q+1)!(N+Q-1)!] equal to (N+Q)/(Q+1) when N,Q>>1 ?? It looks like the Q!/(N+Q-1)! cancels but i don't see how, I am going from my lecturers notes here. Homework Equations The Attempt at a Solution
  22. K

    Inverse factorials. What are they? How to calculate?

    Hi I was trying to solve the equation (m + n)!/m! = v for "n" and found an online calculator http://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?site=quickmath&s1=equations&s2=solve&s3=advanced#reply" which spit out the following as a solution: n = factorial(-1)[vm!] - m I...
  23. R

    Inequality involving two unknowns and factorials

    I need to prove the following: n!/r! >= r^(n-r) With r and n as natural numbers and n>=r I know the LHS will end up being (n-r) terms long as the first r! will cancel out of n! (n>=r), but as they're both unknown, I just left it as 1*2*3*...*(n-2)*(n-1)*n 1*2*3*...*(r-2)*(r-1)*r and I looked...
  24. M

    Solve the Factorials Sum Puzzle: Find the Last Two Digits

    Here is a sum from MATHCOUNTS: What are the last two digits in the sum of the factorials of the first 100 positive integers? From 1! to 4! you can add the units digits, since 5! to ... have 0 in their units place. From that I get 13, and I carry over the 1 over to the next column and...
  25. M

    Calculating Factorials in a Series: Exploring the First Six Terms

    Ok, so here goes nothing I have predict what this is in series form, the factorial in the numerator is really throwing me off. I only have to do the series for the first six terms. C0=C0 C2= -5/2 C0 C4= -3/4 C2 = (5x3)/(2x4) C0 C6= -1/6 C4 = -(5x3x1)/(2x4x6) C0 This is what I have so far...
  26. L

    Factoring (n+1)! - 1 + (n+1)(n+1)! Step-by-Step

    can you please explain (step by step) how to factor the following: (n+1)! - 1 + (n+1)(n+1)! i have the answer, don't know how to get there!
  27. 3

    Factorials & Series - im a little lost here

    I Have a small problem. I know the following: The series: S_{n} = 1 + 2 + 3 + 4 + 5 = \sum_{i=1}^{n=5}i 2S_{n} = 1 + 2 + 3 + 4 + 5 + 1 + 2 + 3 + 4 + 5 = n(n+1) = n^{2} + n Therefor: S_{n} = \frac{2S_{n}}{2} = \frac{n^{2} + n}{2} \sum_{i=1}^{n=5}i = \frac{n^{2} + n}{2} = \frac{25 +...
  28. R

    Why does 0! equal 1 and contradict the concept of multiplication?

    okay...I love math and i enjoy math a lot, mainly for its perfection and the aspect of not contradicting itself. But lately i have been runnig into contradictions and things that should not exist...and they piss me off. The worst of them is solving 0! 0!=1 why? i asked my math teacher...
  29. C

    Proving a perfect square with factorials

    The tex seems to be showing different problems than the ones I'm typing... maybe it's just me, but if what I'm talking about doesn't seem to make any sense, please quote my message to see what I've actually typed in the tex tags. If n is a positive integer and n > 1, prove that nC2 + (n-1)C2...
  30. C

    Efficient Display of Large Factorials in LaTeX

    This is really a TeX/LaTeX question, but I wasn't sure where to put it. (Sorry!) I was looking at "Factoring Factorial n" (Guy, Amer. Math. Monthly Oct. '98) and became interested in looking at large factorials. Before I could really get started manipulating them I wanted a good way to store...
  31. RadiationX

    Simplifying Factorials: General Guidelines for Calculus III

    I need some general guidelines on how to simplify factorials. I'm in Calculus III and the Prof. and unfoutunately our textbook has glossed over how to do this. All the factorials we are dealing with now are in relation to sequences and series. so I'm dealing with expressions that...
  32. D

    Rewriting some Series / Factorials

    Let u = 1 + \frac{x^{3}}{3!} + \frac{x^{6}}{6!} + \frac{x^{9}}{9!} + \dotsb v = x + \frac{x^{4}}{4!} + \frac{x^{7}}{7!} + \frac{x^{10}}{10!} + \dotsb w = \frac{x^{2}}{2!} + \frac{x^{5}}{5!} + \frac{x^{8}}{8!} + \dotsb Show that u^3 + v^3 + w^3 - 3 u v w = 1 Well, here...
  33. M

    Dealing with factorials: Generalization

    The second part of this problem is giving me a hard time. I hope someone can help me with it: An airline pilot reported her itinerary for 7 days. She spent 1 day in Winnipeg, 1 day in Regina, 2 days in Edmonton, and 3 days in Yellowknife. a) how many different itineraries are possible...
  34. F

    Factorials: Simplifying Expressions & Convergence Tests

    I forgot most things about factorials but now I have to use them again to simplify expressions while doing convergence tests... What I know is: (n +1) ! = (n +1) n! (2n +1)! = (2n +1) 2n! ? but I don't know how to deal with these: (n -1) ! = ? (2n-1) ! = ? Thanks for any...
  35. E

    Prime divisors of factorials

    I was reading this tutorial and I came across this part which I didn't quite understand: I don't follow this. What does "a will be counted j times and will contribute i towards t" mean? Why does this show that t=s?
  36. H

    Can Gamma Function be used to Integrate Factorials?

    By definite integral, gamma function can be defined as \Gamma(z)= \int_{0}^{\infty} t^{z-1}e^{-t} dt I've learned some properties of Gamma function but my lecturer didn't tell us the domain of Gamma function. (I'm assuming it is defined for all non-negative real numbers). I thought of...
  37. A

    Factorials, handshake problem, help

    im dealing with a problem that isn't that hard, but its messing me up. its been a long time since i took high school and college math, but I am very smart. im trying to write an equation for the general description of a problem like this: you have a tile with x squares on it. you have a...
  38. P

    Microsoft Excel Factorials Question

    I wasn't really sure what forum this belonged in, but I'm doing an equation with Microsoft Excel, using Factorials. Basically, I'm wondering how you go about using them in an equation. Here's my current equation: =A5!/(A5-B5)!B5! Now obviously that comes back as an error. Does anyone have...
  39. B

    How can I simplify (n+3)!/(n+1)! using the property of factorials?

    How would I go about simplifying something like: (n+3)!/(n+1)! ?
  40. M

    Stirling's approx for large factorials - need trick please

    To calculate the multiplicities of 600 heads in 1000 coin tosses you start with 1000 choose 600 or 1000! / (600! * (1000-600)!) which equals 1000! / (600!) * (400!). Since you can't calculate this easily, apply Stirling's approx. N! = N^N * e^(-N) * sqrt(2piN). Applying this to...
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