Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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$\frac{(n/2)!}{n!} = \frac{1}{2^{n/2}(n-1)!!}$?

I was working on a puzzle involving some rather complex probability when I arrived at two very distinct methods with very different ways of calculating the probability of solving the puzzle. The ...
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2answers
72 views

Find the limit $\,\,\, \lim_{n \to \infty}\Big(\big(1+\frac{1}{n}\big)\big(1+\frac{2}{n}\big) \cdots\big(1+\frac{n}{n}\big)\Big)^{1/n} $

What is the limit of: $$ \lim_{n \to \infty}\bigg(\Big(1+\frac{1}{n}\Big)\Big(1+\frac{2}{n}\Big) \cdots\Big(1+\frac{n}{n}\Big)\bigg)^{1/n}? $$ By computer, I guess the limit is equal to ...
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2answers
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Factorial as a sum. Insight appreciated

I recently posted an answer to a question about ways to express the factorial function as a sum. I posted the following formula, which I discovered several years ago and I haven't seen anywhere else: ...
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0answers
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Simple finite series with reciprocal factorials

I'm trying to find the following sum: $$ \sum_{k=0}^n{1\over(n-k)!}{x^{k+2}\over k+2}. $$ The most obvious way is to differentiate wrt to $x$ leading to $$ ...
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1answer
84 views

A limit with $((n-1)!)^{1/(n-1)}$ and other roots of factorials

How to prove that the following limit is positive? $$ \lim_{n \to \infty}\left(((n-1)!)^{1/(n-1)}-2\left(\frac{((n-1)!)^3}{(2n-2)!}\right)^{1/(n-1)}\right) >0,$$ where $ n\in \mathbb Z, n>1 ...
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7answers
161 views

$\displaystyle\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$

Show that $\,\displaystyle\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$. I'm thinking right now (though not getting anywhere with it) that I want to expand out the summation portion to ...
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1answer
20 views

Finding Factorial using Integral Definition

$n! = \int_{0}^{\infty} {e}^{-x}{x}^{n} \,dx$ How can we find $400!$? $400! = \int_{0}^{\infty} {e}^{-x}{x}^{400} \,dx$ Integration by parts is way too complicated, what are the other options?
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2answers
93 views

Factorial Rational Limit

Anything besides the squeeze theorem. Here it is: $$\lim_{n\to\infty} \frac{(2n - 1)!}{{2n}^{n}}$$ Can someone start me off?
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0answers
24 views

Bounds on constant for Stirling approximation

Stirling's approximation says that $$n!\sim\sqrt{2\pi n}\left(\dfrac{n}{e}\right)^n.$$ What is known about constants $c_1$ and $c_2$ such that $$c_1\sqrt{n}\left(\dfrac{n}{e}\right)^n\le n!\le ...
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1answer
85 views

Show that $n!^{n+1}$ divides $(n^2)!$

My attempt so far is by induction. Let $f(n) = \frac{(n^2)!}{n!^{n+1}}$, I will try showing that $f(n)$ is a positive integer for all $n$. We have $f(0) = \frac{0!}{0!^{n+1}} = 1$. Now assume for ...
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0answers
31 views

Nearest factorial given a number.

Hi suppose I have given a number lets say 344545.Is there a way to determine he nearest smallest factorial? This is a MCQ type question.So what can be the fastest approach.A formal approach is also ...
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1answer
22 views

Calculating the power of prime in factorial by changing base

The greatest power $k$ of a prime $p$ in the prime factorization of $n!$ is equal to $\frac1{p-1}(n-s(n)_p)$, where $s(n)_p$ is the sum of digits of $n$ when represented in base $p$. How to ...
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4answers
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Closed Form for Factorial Sum

I came across this question in some extracurricular problem sets my professor gave me: what is the closed form notation for the following sum: $$S_n = 1\cdot1!+2\cdot2!+ ...+n \cdot n!$$ I tried ...
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1answer
22 views

Trouble understanding factorial algebra

I am having trouble understanding some of the algebraic concepts used here. In fact, the entire thing to me makes sense, except for the second red line. I don't understand how the diagonal swap ...
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0answers
14 views

Integration by parts with Legendre Functions

I need help deriving $\int_{-l}^l [P_l^m(x)]^2 = \frac{2}{2l+1} \frac{(l+m)!}{(l-m)!}$ for the associated Legendre functions I am supposed to use $P_l^m(x) = (-1)^{-m}\int_{-l}^l ...
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1answer
60 views

COmbinatoric : Guess who is the winner candidate?

National Radio Broadcast will put a contest to guess five winners out of twelve local boxers who will compete to win the best 5 boxers. All twelve boxers are equally good so the chance of winning is ...
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3answers
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How to define fractional factorials, like 3.6!? [duplicate]

I did not know that you could find an answer for that. However, I can only use Excel so far to do it. How to calculate 3.6! by hand?
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1answer
728 views

Can a double-factorial be a perfect square?

The title says it, basically. My question is $-$ for $ n \ge 2 $, can $n!!$ be a perfect square, where $!!$ represents the double-factorial? My conjecture is no, but I can't seem to be able to find a ...
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1answer
31 views

Partial Derangements

There are n people and n houses, such that every person owns exactly one distinct house. Out of these n people, k people are special (k<=n). You have to send every person to exactly one house such ...
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1answer
55 views

Summing an infinite series

I have been struggling with a problem involving a Markov Chain. To solve it I need to figure out the following ...
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2answers
45 views

Help with induction proof for recurrent function

I am having issues with the following inductive proof. Prove by induction on $n$ that $$ a(n) = n!\bigg(\frac{1}{0!} + \frac{1}{1!} + \cdots + \frac{1}{(n-1)!}\bigg)$$ for all $n \geq 1,$ where ...
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2answers
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Is there a rule to the terms of a falling factorial?

$\require{cancel}$I discovered that $n!=\xcancel{(n)_{n-1}}n^{\underline{n-1}}=n(n-1)(n-2)\cdots(3)(2)$. I have expanded a few examples: $$2!=\xcancel{(2)_1}2^{\underline{1}}=2\\ ...
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0answers
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Query associated with Factorials and Series

I'm struggling to see how we go from Equation 1 to 2 to 3, which are seen below: Equation 1: $$P(Z=k) = p \binom{k/\delta-1}{n-1}(\lambda\delta)^{n}(1-\lambda\delta)^{k/\delta - n} $$ Equation 2: ...
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2answers
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Summation of Binomial Theorem

The binomial theorem formula: $$\sum\limits_{k=0}^{n} {n \choose k} = \sum\limits_{k=0}^{n}\frac{n!}{k!(n-k)!} = \sum\limits_{k=0}^{n}\frac{n(n-1)(n-2) \cdots (n-k+1)}{k(k-1) \cdots 2\cdot1}.$$ I am ...
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2answers
29 views

Factorials and Mathematical induction

I'm having a bit of trouble understanding mathematical induction, particularly when there's a question with powers or factorials. For example I have a problem 1 x 1! +2 x 2! + 3 x 3! +... + n x n! = ...
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4answers
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Equivalent of $ x(x+1)(x+2)\cdots(x+n)$?

Assume $x>0$. Is there an equivalent for this quantity $$ x(x+1)(x+2)\cdots(x+n)$$ as $n$ tends to $+\infty$? I've tried to write $$x(x+1)(x+2)\cdots(x+n)=x^{n+1}\left(1+\frac ...
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1answer
61 views

Any nice way to find number number of single digit ordered pairs $(a, b)$ such that $a!b! \gt a!+b!$

I have listed them all by brute force : a = 0,1 : no solutions a = 2 : b = 3,4,5,...9 c = 3 : b = 2,3,4...9 I'm wondering if there is a clever approach to ...
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1answer
61 views

How to find asymptotics of this sum

Is there any way to find $f(n)$ in this term: $$\sum_{k=2}^n \frac1{\ln \ln(k!^{k!})} \sim f(n)?$$ The tilde symbol means that $$\lim_{n\to∞} \frac{f(n)}{\sum_{k=2}^n \frac1{\ln \ln(k!^{k!})}} = 1$$ ...
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1answer
15 views

n! mod c where c is a composite number

I am trying to write a program to calculate what is $n! \, \text{mod} \, c$, where $c$ is a composite number. While I understand $a b \, \text{mod} \, c$ is equal to $((a \, \text{mod} \, c) (b \, ...
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1answer
61 views

Finding the asymptotics of $\sum_{k=1}^n a^k k!$? Note that $a > 0$.

There's no way to use integration method in this case. I also tried to use Stolz–Cesàro theorem, but couldn't find right $y_n$. What method should I use?
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0answers
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How are the Stirling-based bounds for the factorial function proven?

According to (26) on wolfram mathworld, one has $$\sqrt{2\hspace{-0.04 in}\cdot \hspace{-0.04 in}\pi} \cdot n^{n+(1/2)} \cdot \operatorname{exp}((-n)\hspace{-0.02 in}+\hspace{-0.02 ...
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0answers
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factorial division word problem

I'm having some problems understanding the following: There are 3 programs being observed 4 times (total of 12 observations). There are 12 people used to investigate these programs, such that they ...
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2answers
585 views

What is the remainder when $1! + 2! + 3! +\cdots+ 1000!$ is divided by $12$?

What is the remainder when $$1! + 2! + 3! +\cdots+ 1000!$$ is divided by $12$. I tried to do it using binomial theorem but that doesn't help. How will we do this? Please help.
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1answer
121 views

Double factorial as a sum

I believe the following equality to hold for all integer $l\geq 1$ $$(2l+1)!2^l\sum_{k=0}^l\frac{(-1)^k(l-k)!}{k!(2l-2k+1)!4^k}=(-1)^l(2l-1)!!$$ (it's correct for at least $l=1,2,3,4$), but cannot ...
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2answers
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Factorials Algebra

I have the following inequality $$5*10^{-10} \geq \dfrac{2^{n+1}}{(n+1)!}$$ Is there any way this can be solved algebraically? If not, is there a method that is better than guessing, for finding the ...
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1answer
23 views

How many straight lines can be drawn between five points (A, B, C, D, and E), no three of which are colinear?

How many straight lines can be drawn between five points (A, B, C, D, and E), no three of which are colinear? Attempt: Given 5 points, a line consist always of 2 points. Thus the total number of ...
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1answer
26 views

How many zip codes are as large as 6000-0000, are even numbers, and have a 7 as their third digit?

When they were first introduced, postal zip codes were five digit numbers, theoretically ranging from $00000$ to $99999$. (In reality, the lowest zip code was $00601$ for San Juan, Puerto Rico; the ...
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3answers
45 views

In how many ways can the word ELEEMOSYNARY be arranged.

In how many ways can be the letters of the word ELEEMOSYNARY be arranged so that the S is always immediately followed by a Y? Attempt: There are 3 Es, and 2 Ys, and and then all letters appear once ...
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1answer
291 views

Why are $\limsup \sqrt[n]{1/n!}=0$ and $\limsup\sqrt[n]{n!}=\infty$?

I was looking at the power series $\sum\frac{z^n}{n!}$ and $\sum n!z^n$, and wanted to compute their radii of convergence. For the first, $\limsup \sqrt[n]{1/n!})=0$, and for the second $\limsup ...
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1answer
21 views

How many different eight note melodies within a single octave can be written if black/white keys alternate.

An octave contains 12 distinct notes(on a piano, five black keys and seven white keys). How many different eight notes melodies within a single octave can be written if the black keys and white keys ...
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2answers
28 views

A coke hand in bridge from deck of cards.

A coke hand in bridge is one where none of the thirteen cards is an an ace or is higher than a 9. What is the probability of being dealt such a hand? Attempt: Suppose the thirteens cards are amoung ...
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1answer
4k views

Last non Zero digit of a Factorial

I ran into a cool trick for last non zero digit of a factorial. This is actually a recurrent relation which states that: If $D(N)$ denotes the last non zero digit of factorial, then ...
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1answer
46 views

What are chances that not all S's will be adjacent given a phrase at random.

IF the letters in the phrase A ROLLING STONE GATHERS NO MOSS are arranged at random, what are the chances that not all the S's will be adjacent. Attempt: Given there are 6 letters that appear twice, ...
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1answer
35 views

Five cards selected out of 52 cards. Find probalbilty sum of the faces is 48 or more.

Five cards are dealt from a standard 52 card deck. What is the probability that the sum of the faces on the five cards is 48 or more? Attempt: Five cards can be selected out of 52 cards by 52_C_5 ...
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1answer
38 views

A bridge hand (13 cards) is dealt from a standard 52 card deck. Given events A and B, find $P(A \cup B)$.

A bridge hand (13 cards) is dealt from a standard 52 card deck. Let A be the event that the hand contains four aces. Let B be the event that the hand contains four kings. Find $P(A \cup B)$. Attempt: ...
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3answers
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Why is $1 \times 3 \times 5 \times \cdots \times (2k-3) = \frac{(2k-2)!}{2^{(k-1)}(k-1)!}$

In order to find out the Catalan numbers from their generating function you have to evaluate the product above. Here is what I thought: \begin{align*} 1 \times 3 \times 5 \times...\times (2k-3) ...
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1answer
56 views

How many ways different sets of values can be chosen for the $x_s$ , if $x_1 + x_2 + x_3 = 20$?

Your statistics teacher announces a twenty-page reading assignment on Monday that is to be finished by Thursday morning. You intend to read the first $x_1$ pages Monday, the next $x_2$ pages ...
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1answer
45 views

How many ways can a twelve member cheerleading be pair up.

Problem: How many ways can a twelve member cheerleading squad(6 men and 6 women) pair up to form 6 male-female teams? What might the number 6!6!2^6 represent? What might the number 6!6!2^6*2^12 ...
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2answers
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Prove $1(1!)+\dots+n(n!) = (n+1)!-1$ using induction

So I'm trying to prove this statement (through induction): $$1(1!)+2(2!)+\dots +n(n!)=(n+1)!-1$$ But I'm confused with the inductive step here: $$(n+1)!-1+[(n+1)(n+1)!] = (n+2)!-1$$ What do I do ...
9
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1answer
179 views

Study of the convergence of a sequence with repeated radicals

Consider the sequence $$ a_n = \sqrt {1!\sqrt {2!\cdots\sqrt {n!} } }, \quad n\in\mathbb N. $$ Does this sequence converge? Clearly, $\{a_n\}_{n\in\mathbb N}$ is monotonically increasing. ...