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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4
votes
3answers
117 views

Evaluating $\lim_{n\to \infty}\frac1{2n}\log\left({2n \choose n}\right)$ [duplicate]

$$\lim_{n\to \infty}\frac1{2n}\log\left({2n \choose n}\right)$$ Now, $\log(n!) = \Theta (n\log(n))$ so I think we could write, $$\lim_{n\to\infty}\frac{1}{2n}\left(\log\left(2n!\right) - ...
-2
votes
1answer
64 views

Determine Even or Odd

Let there a number $n$ and $b$. We need to tell if $n!$ divided by $b$ would be odd or even. How could we determine this problem. Assume if (n! % b)!=0 then answer gets rounded to integer.
8
votes
1answer
139 views

$\frac{x}{10!} = \frac{1}{8!} + \frac{1}{9!}$

I have a pretty simple straightforward question. Q) Find the value of $x$ in the following: $$\frac{x}{10!} = \frac{1}{8!} + \frac{1}{9!}$$ Instinctively, I do the quickest thing I know how to ...
-2
votes
2answers
155 views

Even or Odd for factorial

Moderator Note: This is a current contest question on codechef.com. Given $N$ and $M$ I need to tell whether $\left\lfloor \large\frac{N!}{M} \right\rfloor$ is even or odd.How to do this ...
8
votes
1answer
156 views

Study of the convergence of a sequence with repeated radicals

Let the sequence $$ a_n = \sqrt {1!\sqrt {2!\cdots\sqrt {n!} } } $$ Does this sequence converge? I can tell intuitively that $a_n$ is monotonically increasing. Therefore, there are two ...
0
votes
1answer
83 views

Digit in units place of 1!+2!+…99!

There isn't much I can add to the question description to expand upon the title. I came across this in a multiple choice test. The options were 3, 0, 1 and 7. I am absolutely stumped. Any pointers? By ...
9
votes
4answers
345 views

Evaluate a finite sum with four factorials

Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate $$ \sum^{n}_{i\mathop{=}0}\frac{1}{n+k+i}\cdot\frac{(m+n+i)!}{i!(n-i)!(m+i)!}$$ Any hints? I'm stuck on ...
3
votes
0answers
58 views

Which series of numbers effectively translates the factorial to the exponential function?

We have the relation of the Bernoulli numbers $$B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left(1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\cdots\;\right).$$ For $n>1$, the right hand sum ...
0
votes
1answer
80 views

Falling factorial summands

Representing the summand as a falling factorial Compute the sum $$\sum_{k=1}^n\frac{1}{(k+1)(k+2)}$$
0
votes
2answers
117 views

Trailing zeroes in factorials: are there any excluded values divisible by 5 other than $5$ and $30$?

I've discovered that when this algorithm for counting zeroes on the end of $n!$ is applied to some $n\in\Bbb{N}$: $$f(n)=\sum_{k=1}^{k:n/5^k\le1}\left\lfloor\frac{n}{5^k}\right\rfloor\notin\{5,30\}$$ ...
1
vote
1answer
111 views

Stirling approximation / Gamma function

Is it possible to obtain the Stirling approximation for the factorial by using the gamma function ? $$\Gamma(z) = \lim_{n \to +\infty} \frac{n! n^{z}}{z(z+1) \dots (z+n)}$$ Any hint ?
8
votes
3answers
248 views

Is Ramanujan's approximation for the factorial optimal, or can it be tweaked? (answer below)

Ramanujan's famous factorial approximation, $$n!\approx\sqrt{\pi}\left(\frac{n}{e}\right)^n\root\LARGE{6}\of{8n^3+4n^2+n+\frac{1}{30}}$$ is far more accurate that the Stirling approximation when ...
3
votes
2answers
95 views

Limit of $\frac {n^n}{n!}$ [duplicate]

I have to prove that $$\lim_{n\to \infty} \frac {n^n} {n!}=\infty$$ I've tried to look for a lower bound that also converges to $\infty$ (I don't know if I'm explainig myself correctly), but I ...
3
votes
2answers
112 views

Limit of the sequence $\frac {a^n} {n!}$

I need to prove that $$\lim_{n \rightarrow \infty} \frac {a^n} {n!}=0$$ I have no condition over $a$, just that is a real number. I thought of using L'Hôpital, but it's way too complicated for ...
1
vote
0answers
40 views

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 ...
4
votes
3answers
159 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 ...
2
votes
4answers
475 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. ...
0
votes
1answer
85 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?
1
vote
1answer
88 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 ...
0
votes
1answer
286 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 ...
8
votes
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! = ...
3
votes
3answers
164 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 ...
3
votes
3answers
116 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 ...
3
votes
2answers
120 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?
1
vote
2answers
73 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?
5
votes
1answer
51 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 ...
1
vote
1answer
68 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 ...
5
votes
1answer
297 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 ...
0
votes
5answers
143 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 ...
2
votes
3answers
104 views

Prove properties of the gamma function

$$\sum_{n=1}^{∞} \frac{\Gamma(n)}{\Gamma(n+p+1)}=\frac{1}{p^2\Gamma(p)}$$
2
votes
5answers
248 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.
1
vote
0answers
20 views

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 ...
0
votes
1answer
98 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?
4
votes
2answers
148 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 ...
1
vote
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?
0
votes
2answers
51 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 ...
1
vote
2answers
194 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\\ ...
1
vote
3answers
94 views

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 ...
4
votes
1answer
166 views

More identities of the Ramanujan Double factorial type.

Ramanujan discovered the following identity $$x=\sum_{n=0}^\infty (-1)^n ...
1
vote
1answer
64 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 ...
1
vote
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 ...
0
votes
2answers
64 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 ...
0
votes
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: ...
0
votes
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.
10
votes
4answers
346 views

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 ...
0
votes
1answer
50 views

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

$$^{n}P_{r} = r!$$ Find all possible values of r.
3
votes
1answer
98 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: ...
0
votes
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 ...
1
vote
2answers
200 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 ...
4
votes
0answers
75 views

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 ...