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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Approximating the probability that a range bounds a given number, with very large numbers

Let $m$ numbers be chosen uniformly from $0,\dots,n-1$ without replacement and then sorted in ascending order as $\ell_0,\dots,\ell_{m-1}$. Let there be $b,e,x$ such that $0 \le b \le e \le m$ and $0 ...
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3answers
158 views

How does $\tbinom{4n}{2n}$ relate to $\tbinom{2n}{n}$?

I got this question in my mind when I was working on a solution to factorial recurrence and came up with this recurrence relation: $$(2n)!=\binom{2n}{n}(n!)^2$$ which made me wonder: is there also a ...
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4answers
462 views

Which has a higher order of growth, n! or n^n? [duplicate]

In our algorithms class, my professor insists that n! has a higher order of growth than n^n. This doesn't make sense to me, when I work through what each expression means. ...
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1answer
83 views

Closed form of $n!\sum_{k=3}^{n-1}{{n-2}\choose{k-1}}$

$n$ is given, and it takes part in the following formula. $$n!\sum_{k=3}^{n-1}{{n-2}\choose{k-1}}$$ Is there a nicer way for expressing it? Without the summation sign?
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1answer
87 views

What's a useful recurrence relation for $(2n)!$ in terms of $n!$?

I want to make an algorithm for $n!$ which will divide the number in half and call the algorithm again until n is so close to $0$ that a value of $1$ can be safely returned, and use the value of each ...
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1answer
259 views

Exotic 6-horse race betting probabilities

I'm gearing up for horse racing season, and I'm trying to teach some fellow engineering friends how to bet "exotic" bets by using colored dice to simulate horses. So, the odds for each horse winning ...
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3answers
1k views

Why factorials above 85 contain zero's at the end.

Sorry, I'm not into advanced math, but it wonders me, why factorials above ~85! contain lots of zero's at the end. Example, 100! = ...
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3answers
162 views

Does $n!$ divide $ n^n$?

Today while I was reading on how to shuffle an array I came across a statement that claims we shall not swap an array entry with the whole array range when shuffling the array otherwise we end up with ...
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3answers
99 views

N women and N men. Groups of pairs.

So we have $N$ men and $N$ women. We are creating groups of pairs. It is not necessary to use every man and woman. How many groups can we make ? So if we number them from $1$ to $N$ - let $W_{1}$ be ...
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2answers
119 views

Why does $\sum_{i=0}^{\infty}\frac{a^i}{i!}=e^a$?

Here is a standard identity: $$\sum_{i=0}^{\infty}\frac{a^i}{i!}=e^a$$ Why does it hold true?
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2answers
68 views

Solving an infinite summation involving exponential and factorial

I'm trying to understand an equality that I found in this biology article. $$\sum_{i=0}^\infty\frac{e^{-x}x^i(1-y)^i}{i!} = e^{-x\cdot y}$$ Can you help me proving this equation holds true?
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1answer
50 views

limit of sequence with factorial

How do you show that: $\lim\limits_{n\to \infty} \frac{\left(\frac{n}{2}\right)^{\frac{n}{2}}}{n!}=0$ using the squeeze theorem (I'd like to avoid using Stirling's formula, too). I tried rearranging ...
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1answer
65 views

Find the Perfect Square

I came upon the following question in a recent district math test, and I have no clue how to solve it, besides using a calculator and doing some serious multiplication, but no calculators were ...
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1answer
294 views

Purely combinatorial proof and simplification of identity involving factorials and summations

While trying to decompose factorials into summations, I came up with the following identity $$(n+2)! = 2^{n+1} + \sum\limits_{k=0}^{n-1}\sum\limits_{i=0}^{n-1-k}\sum\limits_{S \subseteq ...
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5answers
137 views

If you toss an even number of coins, what is the probability of 50% head and 50% tail?

If I toss an even number of coins, how can I calculate the probability to obtain head or tail? This question is different from the other because I can fling the coin a different number of times but ...
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3answers
103 views

Prove properties of the gamma function

$$\sum_{n=1}^{∞} \frac{\Gamma(n)}{\Gamma(n+p+1)}=\frac{1}{p^2\Gamma(p)}$$
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5answers
243 views

Limit of $(n-k)! \cdot n^k$ as $n$ approaches infinity

Is it true that $(n-k)! \cdot n^k$ tends to $n!$ as $n \to \infty$? I think it is correct but can't think of a satisfying proof.
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0answers
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The highest power of $(n^r-1)!$ [duplicate]

There is a problem which i tried a lot but didn't reach to any conclusions:How can i show that the highest power of n which is contained in $(n^r-1)!$ is $\frac{n^r-nr+r-1}{n-1}$?My ...
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1answer
84 views

Factorial Length‎

I want to know:For Example:how many digits 10! is without calculate it I need a formula to count for any Integer.Is there formula to calculate the digits of any Integer number?
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2answers
147 views

Why do many calculators evaluate $(-0.5)!$ to $\sqrt\pi$?

According to Wikipedia, factorial only is defined for non-negative integers. How come Spotlight, the Windows calculator and the Google search engine come up with $\sqrt\pi$ if you try to solve ...
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1answer
42 views

Need help at factorial

$$\frac{ (2+x)!}{x!}$$ is there any formula for calculating this? I think : x!(x+2)/x! then results in x+2...is this true?
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2answers
50 views

Combinations from a finite pool of objects

We've got a pool containing 5 A-balls, 4 B-balls, 3 C-balls, 2 D-balls and one E-ball. How many ways are there to pull out 5 balls? I thought of dividing off from the formula: $\frac{15!}{10!}$ but ...
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0answers
44 views

Prove for each $n\in\mathbb{N}$ there exists a prime $q$ such that $n < q < n! + 1$

I'm trying to prove that for each $n\in\mathbb{N}$ there exists a prime $q$ such that $n < q < n!+1$. I think I need to use the fact that every natural number is the product of one or more ...
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2answers
186 views

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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3answers
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An Identity Involving the Pochhammer Symbol

I need help proving the following identity: $$\frac{(6n)!}{(3n)!} = 1728^n \left(\frac{1}{6}\right)_n \left(\frac{1}{2}\right)_n \left(\frac{5}{6}\right)_n.$$ Here, $$(a)_n = a(a + 1)(a + 2) \cdots (a ...
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1answer
158 views

More identities of the Ramanujan Double factorial type.

Ramanujan discovered the following identity $$x=\sum_{n=0}^\infty (-1)^n ...
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1answer
63 views

Binomial coefficient properties

On "theoretical computer science cheat sheet" I found a special formula which is: $$ {n \choose k} = (-1)^k {k-n-1 \choose k}$$ But when I try to expand the value of ${k-n-1 \choose k}$ I have ...
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2answers
65 views

Finding $n$ in $n=ReverseFactorial(x)$ where $x$ is known.

My software application receives a series of very large integers (hundreds of decimal digits). So far, I have been using string/textual representation of decimal digits for very simple manipulation ...
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2answers
62 views

How many zeroes (at the end) will 10! have when written in base 3?

I can get the answer for the question by calculating 10! and then converting it to base 3 but is there a more logical point of view to this question which will mostly generalise the solution for this ...
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1answer
39 views

Summation of a curious series-repeated division by primes

I am interested in knowing if there is some closed form/formula for the following series: ...
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0answers
64 views

The sum of factorials $1!+2!+3!+\ldots n!$ [duplicate]

The sum of factorials of natural numbers, this question was asked in Microsoft's interview at my campus placements today.
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How to prove $\left(\frac{n}{n+1}\right)^{n+1}<\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}<\left(\frac{n}{n+1}\right)^n$

Show that: $$\left(\dfrac{n}{n+1}\right)^{n+1}<\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}<\left(\dfrac{n}{n+1}\right)^n$$ where $n\in \Bbb N^{+}.$ If this inequality can be proved, then we have ...
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1answer
48 views

Find all possible values of r such that: nPr = r! [closed]

$$^{n}P_{r} = r!$$ Find all possible values of r.
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1answer
96 views

Sum of fraction of factorials

Can anybody explain this? $$\sum\limits_{k=1}^{\frac{m-1}2}\frac{(2k)!(2m-2k)!}{(2k-1)(2m-2k-1)k!^2(m-k)!^2}=\frac{(2m)!}{(2 m-1)m!^2}$$ I did actually simplify this to: ...
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3answers
78 views

Is There a Way to Specify Limits On a Factorial

If I want to be able to express a factorial -- let's say "20!" -- but with upper and lower limits such that the factorial is evaluated from Upper Limit, n1=20, through a Lower Limit, n2=10, for ...
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2answers
146 views

Factorials and Prime Factors

I need to write a program to input a number and output it's factorial in the form: $4!=(2^3)(3^1)$ $5!=(2^3)(3^1)(5^1)$ I'm now having trouble trying to figure out how could I take a number and get ...
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0answers
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How to find other nontrivial solutions of $a!b!=c!$? [duplicate]

I know only one nontrivial solution of this equation: $6!\cdot 7!=10!$. There is also a series of trivial solutions: $n!(n!-1)!=(n!)!,\ \forall n\in\mathbb{N}$. So my question is how to find any other ...
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1answer
53 views

Simpler expression for $\sum_{k=1}^{n}{k!}$

Is there a way to express $$\sum_{k=1}^{n}{k!}$$ in a simpler way that doesn't use sums up to n ? I've searched for this around the web and found that the subfactorial function can help with this, ...
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2answers
117 views

Is there a factorial for factorials?

Is there a more succinct way to notate this? $$(n!)((n-1)!)((n-2)!)\cdots(2!)(1!)$$ for clarification, if I had asked a similar question, how to succinctly notate: $$(n)(n-1)(n-2)\cdots(2)(1)$$ I ...
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1answer
292 views

Infinite Sum of factorial denominator and exponential numerator

I've been trying to find the sum of the following infinite series: $$ \sum\limits_{n=1}^\infty \frac{x^n}{n!2^n} $$ I've rewritten it as $$\sum\limits_{n=1}^\infty \frac{y^n}{n!}, y=\frac{x}{2}$$ ...
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1answer
59 views

Limits and common sense

I'm stuck in understanding of limits. It all makes sense, but at a certain point my answers which seem logical to me are not true. Please can somebody explain why as a huge number gets divided by a ...
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1answer
152 views

Factorial of $(7/2)!$ [duplicate]

It's been many years since I studied maths, and I'm trying to figure out the half factorials $(7/2)!$ without a calculator. I did $(7/2) \times (5/2) \times (3/2) \times (1/2) = (105/16) ^ \pi = ...
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1answer
25 views

Number of orders and combinations

I have just done these two questions and I have answers for them but I am not sure if they are correct. A jazz band is to give one concert in each of nine selected cities. Calculate the total ...
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1answer
38 views

Determinant of parametric function and $0!1!2!…n!$

As answer to this question, I trued to calculate the wronskian of: $$\left| \begin{array}{ccc} e^x & e^{2x} & ... & e^{nx}\\ e^x & 2e^{2x} & ...& ne^{nx} \\ e^x & 4e^{2x} ...
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1answer
120 views

What are the conditions for $n^2 \nmid(n-1)!$

Q: What are the conditions for $n^2 \nmid (n-1)!$, given that $2\le n \le 100$ and $n\in \mathbb{N}$? According to me the two conditions must be: 1. $n$ is a prime number (since the factorization ...
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3answers
93 views

Digit sum/product/properties of n!

How would one go about finding the digit sum/product/other properties of n!? If not for n!, at least for n too large for a calculator or computer to compute. (n>1000,let's say). EDIT: People who ...
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1answer
64 views

Where do the two $a!$s come from?

I have \begin{align*} (2a)! &\equiv a! (-a) \dotsm (-3)(-2)(-1) \pmod p\\ & \equiv (-1)^a a!a!\pmod p\\ &\equiv (-1)^a a!^2\pmod p. \end{align*} The $(-1)$ is just to get the ...
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2answers
556 views

How many perfect squares divide 1!2!3!4!5!6!7!8!9!

What I naturally did was to find the prime factorisation of the product of factorials which is $ 2^{30}3^{13}5^5 7^3 $. Clearly there is 15 unique perfect squares that divide $2^{30}$, 6 unique ...
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1answer
45 views

Comparing factorials (From greatest to least)

Let's say we're asked to arrange these factorials in descending order: $1000!, \,\,700!\cdot300!,\,\,500!\cdot500!,\,\,600!\cdot300!\cdot100!$ For the first three, we could do: ...
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6answers
179 views

Proof of $0! = 1$ [duplicate]

I have been recently studying binomial theorem and in that we very frequently encounter factorials. But one queer thing which I found is $0!$. Even more queer is its value which is $0! = 1$. I ...