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.
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 =...
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...
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...
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...
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.
Homework Statement
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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...
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?
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...
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...
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...
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.
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
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...
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...
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.
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...
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...
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...
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...
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 ) ! }\...
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
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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) =...
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.
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...
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...
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.
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...
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!
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...