Questions tagged [factorial]

Questions on the factorial function, $n!=n\cdot(n-1)\cdot...\cdot1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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39 views

How to prove this binomial identity? [closed]

The identity to prove: $$ \binom{2(n+m)}{p} = \left( \binom{n+m}{p} \right)^2. $$ I want to understand the proof but am unable to currently understand how to proceed from beginning to end. Any help ...
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1answer
62 views

A sum that's possibly equal to the Euler-Mascheroni Constant $\sum_{n=1}^\infty \frac{\ln n!}{n^3}$

The following interesting sum seems to approach the Euler-Mascheroni constant $\gamma$. $$\sum_{n=1}^\infty \frac{\ln n!}{n^3} \overset{?}{=} \gamma$$ I've looked at the different ways to express the ...
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53 views

Sum of digits of a factorial

This is question from my college math test for JEE . In this question we are asked to find the missing digits of 19!. I skipped this question in my test but I tried it after the test and I made many ...
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Sum of terms in closed form

I've reduced a problem that I'm doing to showing that for all $n \geq r$ natural numbers, $$\sum_{i=o}^{n-r}\frac{(-1)^i}{(r+i+1) \ i! \ (n-r-i)!} = \frac{r!}{(n+1)!}$$ Or equivalently writing $n = N +...
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72 views

Sum $ \sum_{n=1}^{\infty} {n2^n\over(n+2)!} $?

$$ \sum_{n=1}^{\infty} {n2^n\over(n+2)!} $$ The exercise mentions that this can be written as a telescopic series; I've been trying to write it in such a way but I'm stuck, can't seem to find one! Any ...
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21 views

When are fractions of factorials equal to whole factorials? [duplicate]

10!/7!=6!. 6!/3!=5!. Is there something interesting happening in these examples, or are they just coincidence? Is there a way of finding other examples like these?
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19 views

Premutations of similar objects proof [duplicate]

I'm having a hard time trying to find an intuitive proof of why permutations of n objects including similar objects lets say p, q, r similar objects is $$\frac{n!}{p! q! r!}$$ my question is the same ...
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62 views

Counting the Number of Combinations with Constraints [closed]

Suppose there are 5 balls: Red Blue Green Yellow Orange Normally: There are 5! = 120 ways these balls can be organized (n!) However, I have the following question: Suppose we have the following &...
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Value of series involving central factorial numbers and zeta function

I encountered an interesing series and was wondering if its value can be computed. First we consider the central factorial numbers. For $n\in \mathbb N$ we define the polynomial $$ P_n(x) = x(x + \...
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Comparing Factorials vs 2^n

I have the following question: In general, is $N!$ bigger than $2^N$? Using the R programming language, I made a plot of these $N!$ vs $2^N$: ...
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1000! is divisible by 10^n. Find largest value of n [duplicate]

$1000!$ is divisible by $10^n$. Find the largest value of $n$. This question I found in a maths Olympiad question book. This is a question from the chapter on arithmetic and geometrical progressions.
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Proving $n! = (-1)^n \sum_{i=1}^n (-1)^i\binom{n}{i} i^n $ [closed]

While studying Stirling numbers of the second kind the following formula for $n!$ suddenly popped up! $$n! = (-1)^n \sum_{i=1}^n (-1)^i\binom{n}{i} i^n $$ I was curious if I could find a "direct&...
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1answer
56 views

Solve a factorial equation algebraically or computationally [closed]

Find the $n$ that is the closest solution to the below equation $$ \frac{(4.554 \times 10^{9})!(4.6 \times 10^9 - n)!}{(4.554 \times 10^{9} -n)!(4.6 \times 10^9)!} \approx 0.997 $$ Does anyone have ...
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57 views

Simplifying Products of Factorials

I'm not sure if there's any way to do this, but I figure it's worth a shot. Is there any way to simplify the number of products of factorials into a smaller expression? For example, I have the ...
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2answers
121 views

Is it okay if math questions are ended with exclamation mark (!)? [closed]

I don't know whether there is any rule about this. But I've been wondering that if a math question is ended with an exclamation mark, would it be misinterpreted with "factorial"? For example,...
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306 views

Ways to find $\frac{1}{2\cdot4}+\frac{1\cdot3}{2\cdot4\cdot6}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6\cdot8}+\cdots$

$$\frac{1}{2\cdot4}+\frac{1\cdot3}{2\cdot4\cdot6}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6\cdot8}+\frac{1\cdot3\cdot5\cdot7}{2\cdot4\cdot6\cdot8\cdot10}+\cdots$$ is equal to? My approach: We can see that ...
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28 views

Tricky question about... a infinite reduced row echelon matrix

Let's say I want to write the polynomial function $p(x)$ that passes the points $(1,1!)$, $(2,2!)$, $\ldots$, $(n,n!)$, that will be of the form $p(x) = a_0 + a_1x + a_2x^2 + \ldots + a_nx^n$. What I ...
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3answers
64 views

Showing $f(kx,ky)\geq f(x,y)$

We have a function $$f(x,y)={y\choose x}$$ for $1<x<y$. We are trying to show that for a positive integer $k$, $f(kx,ky)\geq f(x,y)$. So what I did so far was to write $$\frac{ky(ky-1)\cdots(ky-...
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143 views

How do I solve this divisibility problem?

Question 23: Which one of following statements holds true if and only if $n$ is a prime number? $$ \begin{alignat}{2} &\text{(A)} &\quad n|(n-1)!+1 \\ &\text{(B)} &\quad n|(n-1)!-1 \\ ...
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Finding consecutive composite numbers [duplicate]

I was thinking about an elementary question For some given $k\in \mathbb N$, find $k$ consecutive composite integers. The answer that is generally expected is The following $k$ integers are ...
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1answer
52 views

Simplifying sum with rising and falling factorials

Let $(x)^{(n)}=x(x+1)\cdots(x+n-1)$ be the rising factorial and $(x)_{(n)}=x(x-1)\cdots(x-n+1)$ be the falling factorial. I am sure that the sum $$\sum_{n, p, q \geqslant 0}\sum_{ l =0}^{n}\frac{\...
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45 views

Does double subfactorial exist?

Mainly, I have 2 questions regarding factorial. However since I'm only allowed to post one question per one post, I'll post my second question later. The reason I'm asking these questions is because ...
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1answer
61 views

Rising factorials

Let $a^{\bar{n}}$ be a rising factorial or Pochammer symbol and ${\Gamma(a)}$ be the Gamma function. I want to ask if $0^{\bar{0}}$=$1$ or not. As you know there is a relation between the rising ...
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Show that $\sqrt[7]{7!}<\sqrt[8]{8!}$ [duplicate]

I have to show that $$\sqrt[7]{7!}<\sqrt[8]{8!}$$ I've already tried raising both sides to $7\cdot8$ and got $$(7!)^8<(8!)^7$$ and, after a few computations, I obtained $$7!<8^7$$ What should ...
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1answer
43 views

Constant lower bound of this summation involving factorial?

Given $$f(n)=\frac{n}{(2 n) !} \sum_{i=0}^{n-1}\left((n+i-1) ! \sum_{k=0}^{i} \frac{(n-k-1) !}{(i-k) !}\right)$$ where $n$ is natural. Is there a constant lower bound for $f(n)$, i.e., is there a ...
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2answers
50 views

Study the convergence of this: $\sum_n \frac{n! }{ 6\cdot7\cdots(n+5)}$ [closed]

convergence for this one: $$\sum_n \frac{n! }{ 6\cdot7\cdots(n+5)}$$ I tried to calculate it standard and didn't got an answer... Tried D'Alembert but didn't work. Thank you very much for all the ...
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Find all possible values of a natural number $n$ such that $a! + b! = 5^n$

I got the following problem: Find all possible values of a natural number n such that $$ a! + b! = 5^n $$ My solution: Since $5^n$ is always an odd number, a! needs to be odd and b! to be even or ...
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3answers
91 views

Inequality $(n!)^3<n^n(n+1/2)^{2n}$

$(n!)^3<n^n((n+1)/2)^{2n}$ I've been trying to solve this problem. I've tried using these relations: $n^2(n+1/2)^2=$Sum of cubes of first n natural numbers $(n(n+1)/2)=$Sum of first n natural ...
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215 views

Primes made from alternating factorials

While messing around with factorials, I noticed this: $$3! - 2! + 1! = 6 - 2 + 1 = 5$$ $$4! - 3! + 2! - 1!= 24 - 6 + 2 - 1=19$$ $$5! - 4! + 3! - 2! + 1! = 5! - 19 = 101$$ $$6! - 5! + 4! - 3! + 2! - 1! ...
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1answer
67 views

What is $(3n+3)!$ equal to

Is $(3n+3)!$ equal to: a) $(3n+3)\cdot(3n)\cdot(3n-3)\cdot(3n-6)\cdot...\cdot(1)$ b) $(3n+3)\cdot(3n+2)\cdot(3n+1)\cdot(3n)\cdot...\dot(1)$ I was wondering, since $$(n+1)!=(n+1)\cdot n!$$ Shouldn't ...
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'Provide the answer as a function of N. You are not required to show the derivation process.' What should the answer be?

The question(as an example) I have N cities and I need to find out how many paths does the circuit have. The problem For an assignment my teacher asked me to provide the answer as a function of N. ...
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51 views

Prove that $\sum_{q=0}^{d-r}\sum_{s=r+q}^{d}{{\binom {r-1+q}{r-1}}(r-1)!s}=\sum_{s=r}^{d}{\binom{s}{r}(r-1)!s}$ by sum manipulation [closed]

I know the sums $$\sum_{q=0}^{d-r}\sum_{s=r+q}^{d}{{\binom {r-1+q}{r-1}}(r-1)!s}$$ and $$\sum_{s=r}^{d}{\binom{s}{r}(r-1)!s}$$ are equal. How do I manipulate the double sum to look exactly like the ...
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1answer
48 views

Prove that an equation with two integer variables has no solution [closed]

$$\dfrac{(2n)!}{((6k)n!(2^n)+1)} = n^2$$ How will you prove that it has no solution if $n$ and $k$ are integers?
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Is this function increasing in $x$?

Say we have $f(x)=\frac{x!}{(x-n)!n!}$ for any $n\geq1$ and integer and $x>n$ and integer. How can we show whether it's increasing or decreasing in $x$? I was thinking, if $x_1\leq x_1$, then $x_1!\...
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148 views

Can $n!$ be expressed as the sum of $n$ powers of 2?

New here, so this question may be a little bit messy. Also, I don't know if this has been asked yet, so sorry if this is a duplicate of another question. Yesterday, I received the following question: ...
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2answers
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Is $\frac{y!}{y+1}$ an integer if $y$ is an odd number? [duplicate]

I was playing around with equations and I somehow stumbled upon the idea that if $y$ is an odd number then $\dfrac{y!}{y+1}$ is an integer. I have tried many numbers, but I do not know if I can prove ...
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1answer
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$ \sum_{x=1}^n {{n-1} \choose {x-1}} {N \choose x} = {{N+n-1}\choose n}$ [duplicate]

I am working on a random variable problem. To prove a PMF to be a valid one, I need to establish the following identity : $ \sum_{x=1}^n {{n-1} \choose {x-1}} {N \choose x} = {{N+n-1}\choose n}$ I ...
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1answer
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How many ways are there to choose an arbitrary number of students (including the possibility of choosing 0 students) from 6 students? [closed]

I'm a little confused by the wording of this question, more specifically "an arbitrary number". Is the answer to this 6!? Or simply 7? Thanks!
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Solve $(n+1)^k-1=n!$ [duplicate]

Solve $(n+1)^k-1=n!$ in positive integers. So far, I can see that $n! = -1$ (mod $n+1$), which is true for all $n+1$ prime. Also, $(1,1), (2,1), (4,2)$ are solutions. They seem like special cases, and ...
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2answers
59 views

Evaluating $\sum_{i=0}^{H-1} \dfrac{(-1)^i}{i! (H-1-i)! (k+i)}$

I've been struggling to directly find a closed form to the following sum $$ \sum_{i=0}^{H-1} \dfrac{(-1)^i}{i! (H-1-i)! (K+i)}\,,\qquad K,H\in\mathbb{N}-\{0\}\,. $$ I've found the result indirectly by ...
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1answer
214 views

Identity involving double sum with factorials

In the course of a calculation, I have met a complicated identity, which I want to prove. Let $m>0$ and $0<\ell<m$ be integers. Let $(x)^{(n)}=x(x+1)\cdots (x+n-1)$ be the rising factorial. ...
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1answer
39 views

Show that $K^{(m)}(1)$ is equal to the $m$th factorial moment.

$\textbf{1.98.}$ Let $X$ be a random variable such that $K(t)=E(t^X)$ exists for all real values of $t$ in a certain open interval that includes the point $t=1$. Show that $K^{(m)}(1)$ is equal to ...
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26 views

Number of ways to create a binary string of length on $N$ with no consecutive $1'$s [duplicate]

I stumbled upon a question today and came up with a solution inspired by this article. The question statement is: In a mathematics class, Teacher ask Alice to find the number of all n digit distinct ...
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2answers
270 views

$\infty! = \sqrt{2\pi}$?

The YouTube video "Infinity Factorial" by BiBenBap says that $\infty! = \sqrt{2\pi}$. First of all, isn't the factorial operator an arithmetic operator? It is just multiplication following ...
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1answer
44 views

Inequality with falling factorials

in The Art of Computer Programming from Donald Knuth I came across the following inequality: \begin{equation} N^k - \binom{k}{2}N^{k-1} \leq N^{\underline{k}} \leq N^k. \end{equation} where $ N^{\...
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210 views

Is 7 the only solution for $n!+1 \equiv 0 \mod 10n+1, n \in \mathbb{Z}^{+} $?

I've tested each $0<n<10^5$, it seems that 7 is the only solution for $n!+1 \equiv 0 \mod 10n+1$. But I can't prove it correct or wrong. I know that no solution exists if $10n+1$ is composite, ...
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0answers
17 views

Evaluate $\lim_{x \to 0} (x!)^\frac{1}{x}$ [duplicate]

I looked at the Desmos graph of: $$(x!)^{\frac{1}{x}}$$ I was surprised to find out the limit at 0 wasn't something trivial like 1 or 0, but was instead a number slightly smaller than ln(2). I do not ...
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42 views

The product of $n$ consecutive integers is divisible by $n!$, using induction in conjunction with $\binom{n}{r}=\binom{n-1}{r}+\binom{n-1}{r-1}$

I have seen this proof which uses induction. It seems to be a way of proving the property $\binom{n}{r}=\binom{n-1}{r}+\binom{n-1}{r-1}$, instead of using this in the induction. I have tried to start ...
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22 views

Factorial Function (Decreasing by 2)

If the factorial function is n*(n-1)(n-2)... Then what would the function n(n-2)*(n-4)... be?
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32 views

The equation $\,m^2=1+\sum_{k=0}^{l-1}{l\choose k}(n-k)!\,$ as a generalization of Brocard's problem

Let's consider the equation in integer values $(l,m,n)$ $$m^2=1+\sum_{k=0}^{l-1}{l\choose k}(n-k)!$$ If $\,l=1\,$ the equation is equivalent to Brocard's problem $$m^2=1+n!$$ whose only solutions $(m,...

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