# 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. ### I Relationship between factorials and squares of natural numbers

Was fooling around and wrote down these two equations today that appear to work. I'm not all that bright and I'm positive these either have some proof or restate some conjecture--probably something in a textbook. Could somebody help me out? \forall n \in \mathbb{N}_0\smallsetminus\{0\} n^2 =...
2. ### A Can falling factorials be a Schauder basis for formal power series?

We usually talk about ##F[[x]]##, the set of formal power series with coefficients in ##F##, as a topological ring. But we can also view it as a topological vector space over ##F## where ##F## is endowed with the discrete topology. And viewed in this way, ##\{x^n:n\in\mathbb{N}\}## is a...
3. ### MHB Factorials and Exponent Challenge

Find all positive integer solutions $(a,\,b,\,c,\,n)$ of the equation $2^n=a!+b!+c!$.
4. ### I Intersections between this infinite power tower and a multifactorial

For those unaware of multifactorial notation, it should be noted that there are some common mistakes made when first being introduced to the notation. For example, ##n! \neq (n!)!## and ##n! \neq (n!)! \neq (n!)! \neq ((n!)!)!##. Just to make sure we're all up to speed, here's a quick run down...
5. ### 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...
6. ### Algebraic manipulation with factorials

I substituted and got ((xn/2nn!) + 1)/(xn/2nn!). I then multiplied by 2nn! to each side and got (xn + 2nn!)/(xn). Now I am confused as to what my next step should be.
7. ### Permutations (with repetitions) problem

Homework Statement [/B] The question is phrased in the following way: There are 6 jobs to be assigned to 5 people. Each job is assigned only to one person, and each person must have at least one job. How many different arrangements are there? Homework Equations In general, I would approach a...
8. ### 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?
9. ### I What is the Notation for Factorials?

I just have a quick question on how to write the notation for a factorial. I have a series with a factorial of 5*10*15*...*(5n) in it. Is this written as 5n!, as (5n)!, or something else? I'm pretty sure it's 5n!, as I've written 5n! out as 5(1*2*3*4*...*n), which when you distribute the 5...
10. ### A question about factorials, nCr

Homework Statement Hi,I found this problem when I was reading a book,I knew the answer but there is one thing that I don't understand.Here is the question: Given C(2015,m),find the smaller of m such that C(2015,m) is an even number Homework EquationsThe Attempt at a Solution Here is the...
11. ### Testing primes using factorials

As far as I can tell for the equation n!/n^2=x or n-1!/n=x, if x is a natural number then it seems n is composite. If x is a non-natural number then it is prime (excluding 4). I am aware that this is not very practical since I am using factorials and the numbers get very large. But it still...
12. ### 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.
13. ### 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
14. ### How to cancel factorials in power series problems?

I have been practicing power series problems and a lot of them include factorials. To find out if they converge or not I'll often use the ratio test. However, I never quite understood how to cancel factorials when replacing the n with n+1. i.e. the textbook has an example problem that shows that...
15. ### MHB Modular arithmetic and factorials

Hi there, I actually have a few questions I came across on my studies. They are (a) Show that if p is odd and x is an integer such that x^2 ≡ 1 mod p^k, then x = ±1 mod p^k (b) Find the solutions of the congruence equation x^2 ≡ 1 mod 2^k (c) What is the remainder of (p − 1)!, when divided by...
16. ### Factorials within alternating series

Homework Statement ∑ [ (-1)^n * n!/(10^n) ] 2. The attempt at a solution the problem is that I cannot use derivative to make sure that a(n) is decreasing neither L hopital rule to find the limit.
17. ### Express (2n+1)(2n+3)(2n+5) (4n-3)(4n-1) in factorials

Homework Statement Express (2n+1)(2n+3)(2n+5)...(4n-3)(4n-1) in terms of factorials Homework Equations n!=n(n-1)! The Attempt at a Solution I know (2n+1)+(2n+3)+⋯+(4n−1)=∑2n−1+2k, where k starts as k = 1 and increases to infinity. Then I was stuck. I am trying to learn maths on my own but it...
18. ### Sequence (n)/(n^n) Convergent or Divergent and Limit?

Homework Statement Is the sequence {(n!)/(n^n)} convergent or divergent. If it is convergent, find its limit. Homework Equations Usually with sequences, you just take the limit and if the limit isn't infinity, it converges... That doesn't really work here. I know I'm supposed to write out the...
19. ### Why is There No Inverse Factorial Function?

Why exactly is there no such thing as an inverse factorial function? Now I am fully aware of the fact that the factorial function (##f(x) = x!##) is not one-to-one, since both 0! and 1! equal 1. But can't we circumvent this by restricting the domain of f such that it only includes values of x...
20. ### MHB Integrals at infinity/ factorials problem

Need help on exercise 2 from the linked image , left first in so you guys could see the Γ(χ) function any help is appreciated , thanks in advance!
21. ### The Square Root Function: Understanding the Difference

I went to splash at MIT a while back, and I took a class on cesaro summation. We were promised to go over an interesting identity but we never did: ##4(\frac{1}{2}!)^2=\pi##. Now, this doesn't make any sense to me, since I thought you could only do factorials with integers, like in the famous...
22. ### Can factorials be integrated in this equation?

Hello, well here's my problem: I got this integral and I don't know how to calculate it (I am trying to find if there exists a k that satisfies this relation) : \int_0^k \frac{1}{ ( 4k-4r-2 ) ! ( 4r+1 ) ! }\, \left ( \frac{y}{x} \right )^{4r} dk = \int_0^k \frac{1}{ ( 4k-4r ) ! ( 4r+3 ) ! }\...
23. ### MHB Factorials as n-- > infinity?

A little confused here.. what does \frac{(2n-1)!}{2n!} as n---> \infty = to? How would I look at that by inspection and figure it out because I am confused.. isn't 2! = 1 * 2 and 3! = 1 * 2 * 3? but what is 2n! factorial... Also What is 2n-1! factorial equal to
24. ### Fractional Factorials (Statistics). Design Identification

Homework Statement Anand, Bhadkamkar, and Moghe (1995) used a fractional factorial design to determine which of the six possible factors influenced the determination of manganese in cast iron. The six factors and their levels follow: A-Titration speed ; Medium...
25. ### Binomial Coefficient - Factorials Part III

Homework Statement ##| \ X \in \mathcal P(\{0,1,2,3,4,5,6,7,8,9\}) : |X|= 4 \ | = \ \ ?## Homework Equations There's no wording in the exercise , just what I wrote above.If I understood correctly , they asked me to find the cardinality of the set of all subsets of {0,1,2,3,4,5,6,7,8,9} that...
26. ### Binomial Coefficient - Factorials Part II

Homework Statement This one is trickier than the problem in my other thread in my opinion.Twenty-one people are to be divided into two teams , The Red Team and the Blue Team.There will be 10 people on Red Team and 11 people on Blue Team.How many ways to do this? I am not sure how to solve...
27. ### Set Theory - Counting - Binomial Coefficient - Factorials

Homework Statement A department consists of 5 men and 7 women.From this department you select a committee with 3 men and 2 women.In how many ways can you do this? Homework Equations Since the "overall set" (the entire department) is composed of both men and women and each has a specific...
28. ### Factorials and lists/subsets counting

1.1 Homework Statement Using only pencil and paper , find the value of ##\frac{120!}{118!}## 2.1 Relevant equations ##\frac{120 \cdot 119 \cdot 118!}{118!} = 120 \cdot 119 = 14280## 1.2 Homework Statement Compute how many 9-digit numbers can be made from the digits...
29. ### What is the pattern in factorials and squares?

Hi there, I don't really have a question but I just thought I'd share something that I've found and see if anyone could make any sense of it, or find some sort of pattern in the results. I noticed that for some of the first few factorials (from 4! to 12!), (ceiling[(n!)0.5]2-n!)=a perfect...
30. ### Rate of growth of factorials.

I was wondering how fast x! grows as it approaches infinity as compared to x2, 2x, and xx. The last one was fairly obvious since x*x*x*x... > x(x-1)(x-2)(x-3)... But I can't figure out a way to show that x! grows faster than x2 or 2x. I know it grows faster since I can compare the graphs of...
31. ### MHB What is the Sum of Factorials for a Specific Equation?

Evaluate \frac{2^2-2}{2!}+\frac{3^2-2}{3!}+\frac{4^2-2}{4!}+\cdots+\frac{2012^2-2}{2012!}
32. ### What is the Relationship Between Factorials and Unit Digits?

Homework Statement Let ##100!=N\cdot 10^n##. If N is relatively prime with 10 and unit digit of N is d, then n+d is equal to A)26 B)28 C)30 D)32 Homework Equations The Attempt at a Solution I don't think it would be a good idea to expand the factorial and separately write out...
33. ### Understanding Factorials and Multiplying by an Integer

Homework Statement Hey all. Not super familiar with using factorials, however, they do pop up occasionally. I understand that n! = 1*2*3*...*n. How do we treat factorial when we are multiplying n by an integer before taking the factorial? I know the answer for expanding (2n)!, however, I do...
34. ### Horrible limit with factorials. Need to use Stirling formula?

Homework Statement Homework Equations The Attempt at a Solution Do I need to used the boring Stirling formula?
35. ### Exploring Solutions to a!b! = a! + b! + c^2 for Positive Integers a, b, and c

Given that a, b, and c are positive integers solve the following equation. a!b! = a! + b! + c^2 anyone?
36. ### Matlab: factorials without for loops or colon

The Problem: Write a function that finds the factorial of a positive integer without using for or while loops, the factorial function, or the : range operator. Honestly, I don't really know how to start with this one. If I were able to use a for loop it would be easy, and I don't see how I...
37. ### Infinite Series using Falling Factorials

Homework Statement Determine \sum_{k=0}^\infty \frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)}. Homework Equations ## (x)_m=x(x-1)(x-2)...(x-(m-1)) ##, integer ##m\geq0##. ## (x)_{-m}=\frac{1}{(x+1)(x+2)...(x+m)}##, integer ##m>0##. ## Δ((x)_m)=m(x)_{m-1}## \sum_{a\leq k<b}...
38. ### Simplifying Factorials: Proving (n+1)(n+1)!+(n+1)! =(n+2)!

Homework Statement (n+1)(n+1)!+(n+1)! =(n+2)! simplify The Attempt at a Solution I need to know how to simplify this to show it is true. I know that the above statement is true, but I do not understand how to simplify the left hand side to show it. Thanks, I really have no idea where...
39. ### I have troubles simplifying this quotient of factorials

Homework Statement I'm trying to self-study Mary L. Boas' book Mathematical Methods in the Physical Sciences. One of the exercices asks the reader to find the limit of n -> ∞ (n!)2 / (2n)! Homework Equations None The Attempt at a Solution Instinctively I know that (2n)! grows...
40. ### Fortran Understanding Fortran Factorials in Infinite Sums

I don't understand at all how you tell the computer to evaluate a complicated factorial expression such as the one given in in the infinite sum of binomial theorem as Ʃ [n! / k!(n-k)! ] * x^k where n is the final value of the sum and k is where you are in the loop. It's supposed...
41. ### Expressing $\Gamma$(n+$\frac{1}{2}$) for n $\in$ $\mathbb{Z}$ in Factorials

Homework Statement Express \Gamma (n+\frac{1}{2}) for n\in\mathbb{Z} in terms of factorials (separately for positive and negative n). Homework Equations The Attempt at a Solution I've got for n\geqslant 0 that \displaystyle \Gamma \left(n+\frac{1}{2} \right) =...
42. ### Finding the Limit of a Sum with Factorials

Homework Statement Find the limit lim_{n \to \infty} \sum_{j=1}^n \frac{b^j}{(j+1)!} Homework Equations Geometric series sum: S=\sum_{j=1}^n r^n S-rS=(1-r)S=1-r^{n+1} S=\frac{1-r^{n+1}}{1-r} S \to \frac{1}{1-r} \,\,\, as \,\,\, n \to \infty if...
43. ### How to Simplify Factorials: A Step-by-Step Guide

How can ((n+1)^2(*n!))/((n+1)!*n^2) be simplified to (n+1)/n^2? My own answer is (n+1)^2/n^2, but its apparently wrong
44. ### Computing end-digits of large factorials

The factorial of 1 trillion ends in many trailing zeros. Find the five digits that comes before the trailing zeros. I know how to calculate the number of trailing zeros, but don't know what to do afterwards. This is a computational problem.
45. ### Solving an equation involving factorials

Splitting the reciprocal of a factorial into a sum of reciprocals of positive integer I'm interested in finding all positive integers x, y such that {1 \over x} + {1 \over y} = {1 \over N!}, N \in \mathbb{N}. I think it's best to gather as many properties of solutions as possible, to make this...
46. ### Proof of equality involving factorials

Hello everyone Homework Statement i don't see a connection why Cin = \frac{(n*(n-1)*(n-2)*...*(n-i+1))}{1*2*...*i } = \frac{n!}{i!(n-i)!} Is there a way to simplify them in order to see why the equality holds? The Attempt at a Solution the definition of factorial being n!=1*2*...*n I...
47. ### Is there a new way to calculate derivatives of factorials?

I think I have found a formula for finding Successive derivatives of factorials, though it may have been found already. I have attached it to this post.
48. ### Alternating Series involving factorials

I have a specific problem but more than figuring out the answer I just want to figure out how to deal with factorials. My book is less than helpful on it... The problem is... \sum_{n=1}^{\infty} (-1)^n \frac{n^n}{n!} I understand that I have to take the limit of the sequence...
49. ### Number Theory: Division with remainder of factorials

I'm struggling with how to even begin with this problem. Find the remainder of the division of 75!*130! by 211. 211 is prime, so I know the remainder is not 0. I'm not sure where to start though. Thanks!
50. ### Powers of integers and factorials

I would like some direction on studying powers of integers and if they are in any way related to factorials. I was studying the sequence of cubics 1, 8, 27, 64, 125 and so. After a certain number of rounds of a basic rule I choose to apply to this sequence, I arrived at a new sequence...