# Tagged Questions

46 views

### Simplify factorials into a combinatorial formula

Is there any way to simplify this into a combinatorial formula? $$\frac{t!(n-t)!}{n!}$$
47 views

### Given $n$, find smallest number $m$ such that $m!$ ends with $n$ zeros

I got this question as a programming exercise. I first thought it was rather trivial, and that $m = 5n$ because the number of trailing zeroes are given by the number of factors of 5 in $m!$ (and ...
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### An identity for the product of even numbers (double factorial)

I'm unable to prove this identity: Prove that: $2\cdot 4 \cdot 6 \cdot 8 \cdots 2n = 2^n \cdot n!$ Wouldnt it be like this? $2(1 \cdot 2\cdot 3\cdot 4 \cdots n)= 2 \cdot n!$
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### 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 ...
47 views

### help on manipulating this algebraic expression

So I have something like: $\frac {k!}{(k-3)!3!}$ I'm going to add $\frac 12k(k-1)$ to this, and I want to obtain $\frac {(k+1)!}{(k-2)!3!}$ as the result. I'm having trouble with this since I need ...
194 views

### How do you solve an inequality with the factorial of a variable?

How do you solve an inequality with the factorial of a variable? Example: Determine the interval of $n \in \Bbb N$ for which the following inequality holds: $$n! \leq 157788 \cdot 10^{10}$$ Can ...
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### How $\alpha(\alpha+1)\ldots(\alpha+k-1)=\frac{\Gamma(\alpha+k)}{\Gamma(\alpha)}$?

Probability function of Negative Binomial Distribution, $NB(\alpha,p)$, is $$P(X=k)=\binom{\alpha+k-1}{k}(1-p)^{\alpha}p^k,\quad \alpha>0$$ Probability generating function of Negative Binomial ...
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### Simlifying [(k+1)! - 1] + (k+1)((k + 1)!)

I'm afraid I've gotten a bit rusty on Math since I was last in university. I was looking at a problem in my text book that simplified ...
32 views

### Simplify the following problem

How $$\frac{1}{k}\sum_{m=r}^{k-1}\frac{m!}{(m-r)!}=\frac{(k-1)(k-2)\ldots(k-r)}{r+1};\quad r=1,2,\ldots$$ I have thought in two ways: ...
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### Simple Calculation

How $$\sum_{m=r}^{k-1}m(m-1)\ldots(m-r+1)=\sum_{m=r}^{k-1}\frac{m!}{(m-r)!};\quad r=1,2,\ldots$$ ? i have ...
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### Show that $\frac{1}{r!}-\frac{1}{(r+1)!}\equiv\frac{r}{(r+1)!}$.

Show that $\frac{1}{r!}-\frac{1}{(r+1)!}\equiv\frac{r}{(r+1)!}$. I get $$\frac{1}{r!}-\frac{1}{(r+1)!}=\frac{(r+1)-r!}{r!(r+1)!}$$ and in the numerator since $$(r+1)!-r!=r$$ so ...
262 views

### Dividing factorials is always integer

Is there a simple way to show that $$n!\over r!(n-r)!$$ is always an integer?
6k views

### What are the rules for factorial manipulation?

I know that $$(k+1)! - 1 + (k+1)(k+1)! = (k+2)! - 1$$ thanks to wolframalpha, but I don't understand the steps for simplification, and I can't seem to find any rules about factorial manipulations ...
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### Proving with factorials

Let x and y be the postive integers. Show that : $\displaystyle\frac{(x + y)!}{ (x + y)^{(x + y)}} < \frac{x! y!}{ (x^x + y^y)}$ Are there any identities we can use to easily prove this?
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### Number of solutions for $\frac{1}{X} + \frac{1}{Y} = \frac{1}{N!}$ where $1 \leq N \leq 10^6$

Note: this is a programming challenge at this site For this equation $$\frac{1}{X} + \frac{1}{Y} = \frac{1}{N!}\quad ( N \text{ factorial} ),$$ find the number of positive integral solutions for ...
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### Factorial Identity - True or False?

Let $x$ and $y$ be positive integers. Then, is \begin{align} \frac{x^{xy}}{(xy)!} = \sum_{k_1+...+k_x = xy} \frac{1}{(k_1)!...(k_x)!} \end{align} true, where $k_1$, ..., $k_x$ are all positive ...
905 views

### Simplify a factorial

I have the problem to evaluate the following: $$(2n)!\over 2^n(n!)$$ Does this reduce to anything in particular? I stuck it into a computer and it's ...
3k views

### Highest power of a prime $p$ dividing $N!$

How does one find the highest power of a prime $p$ that divides $N!$ and other related products? Related question: How many zeros are there at the end of $N!$? This is being done to reduce ...
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### $n^s=(n)_s+f(s)$, what is $f(s)$?

In the following equation, $$n^s=(n)_s+f(s)$$ What is general form for $f(s)$? Understand that, $$(n)_s=n(n-1)(n-2)\cdots(n-[s-1])=\text{ The Falling Factorial }$$ I have experimented with this ...
172 views

### What makes $0!$ equal to 1? [duplicate]

Possible Duplicate: Prove $0! = 1$ from first principles I don't understand how it's equal to 1. Also, I found that $(-x)!$ is equal to complex $\infty$. How is this so?
181 views

### Simple factorial

I'm re-learning factorials, and I encountered this exercice, but the solution had a diferent result than I got, and no matter how much I try to search, I can't find an explanation to the last step of ...
247 views

### Help with factorials, permutations, and combinations

I am an eighth grader in need of some help. I was assigned a school project on making a java application that computes the total permutations of to given numbers where nPr and later on nCr. I ...
173 views

### Prove $0! = 1$ from first principles
How can I prove from first principles that $0!$ is equal to $1$?
### Number of zeros not possible in $n!$ [duplicate]
Possible Duplicate: How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes? The number of zeros which are not possible at the end of the $n!$ is: ...