# Tagged Questions

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### 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 ...
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### 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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### How do these equate?

I need to evaluate the following $$\frac{(n+1)!}{(n+1)^{(n+1)}} * \frac{n^n}{n!}$$ It should come to $$(\frac{n}{n+1})^n$$ Currently, I only know that the $(n+1)!$ cancels with the $n!$ to make ...
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### Problem finding limit - which function is asymptotically larger

I have a homework question, so please don't answer fully but I would appreciate a push in the right direction. Basically we need to figure out if $n^{n+\frac{1}{2}}e^{-n}$ is larger,smaller, or equal ...
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### Combinatorics - find $n!$ using inclusion-exclusion [duplicate]

difficult question I need help with. We are asked to show that $n! = \sum_{k=0}^n (-1)^k\binom{n}{k}(n-k)^n$ There is also a hint "try to think of the number of permutations of n elements using ...
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### Use a factorial argument to show that $C(2n,n+1)+C(2n,n)=\frac{1}{2}C(2n+2,n+1)$

I need some help, showing that the left hand side is equivalent to the right hand side. I tried but I get stuck, I am not sure if I am on the right path. Here is my attempt: $C(2n,n+1) + C(2n,n)$ ...
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### Homework - algebra, find constants

The question is as follows, I think I solved it partially: Show that there are $a,b$ real positive numbers such that $an^7 \leq \frac{n!}{7!(n-7)!} \leq bn^7$ $7\leq n$ my solution for b: ...
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### Trying to understand an exercise using factorials with induction

Exercise: Prove that (n + 1)! - n! = n(n!) for any n $\ge$ 1 Given Answer: I will skip the basic step since I understand that part. (n + 2)! - (n + 1)! = (n + 1)!(n + 2) - n!(n + 1) I understand ...
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### Derivative of $\frac{e^x}{x!}$

I am having a bit of trouble putting all the differentiation rules together with the following problem: $$\frac{d}{da} \Bigg(\frac{a^x}{x!}e^{-a}\Bigg)$$ Where $x$ is a discrete variable and $a$ is ...
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### Limit of series involving ratio of two factorials

$$\sum^{\infty}_{j=0} \frac{(j!)^2}{(2j)!} = \frac{2 \pi \sqrt{3}}{27}+\frac{4}{3}$$ The above series is in a homework sheet. We're not expected to find the limit, just prove its convergence. ...
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### Does $a!b!$ always divide $(a+b)!$

Hello the question is as stated above and is given to us in the context of group theory, specifically under the heading of isomorphism and products. I would write down what I have tried so far but I ...
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### Colored Blocks Factorial

I was given the following problem: A bag contains 14 red blocks, 10 white blocks and 12 blue blocks. How many different 19 block sets can be created from the collection in the bag? Two block sets ...
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### Two questions on finding trailing digits in (large) numbers and one on divisibility

Without using a calculator, how can we solve the following? How do we find the number of zeros at the end of $600!$ What are the last 3-digits of $171^{172}$? What is the sum of all positive numbers ...
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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 ...
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### Prove that for any $x \in \mathbb N$ such that $x<n!$ is the sum of at most $n$ distinct divisors of $n!$

Prove that for any $x \in \mathbb N$ such that $x<n!$ is the sum of at most $n$ distinct divisors of $n!$.
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### How to prove that $\mathrm{Fibonacci}(n) \leq n!$, for $n\geq 0$

I am trying to prove it by induction, but I'm stuck $$\mathrm{fib}(0) = 0 < 0! = 1;$$ $$\mathrm{fib}(1) = 1 = 1! = 1;$$ Base case n = 2, $$\mathrm{fib}(2) = 1 < 2! = 2;$$ Inductive case ...
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### Smallest positive integer

Let S(m) is the sum of the factorials of the digits of integer m. I try to find the smallest positive integer n with S(n)=111. My answer is 12334444. Is it right?
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### Determining the type of convergence of a series

The specific homework problem is $\displaystyle\sum\limits_{n=0}^\infty \frac{2\cdot 4\cdot 6\cdots2n}{n!}$. It is problem #29 from the section 11.6 exercises (pg. 737) from Single Variable Calculus: ...
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### Finding all the numbers that fit $x! + y! = z!$

I have the formula $x! + y! = z!$ and I'm looking for positive integers that make it true. Upon inspection it seems that x = y = 1 and z = 2 is the only solution. The problem is how to show it. ...
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### How to find $\beta$ and $\alpha$?

$\mathbb{P}$ is the prime numbers set. $p \in \mathbb{P}$ $a,b,c \in \mathbb{N}$ $n=a p^b+c$ where $c= n\bmod p$ $b$ is the highest power of $p$ who divides $n-c$ How to find $\beta$ where ...
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### Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.

Why is $$\lim_{n \to \infty} \frac{2^n}{n!}=0\text{ ?}$$ Can we generalize it to any exponent $x \in \Bbb R$? This is to say, is $$\lim_{n \to \infty} \frac{x^n}{n!}=0\text{ ?}$$ This is ...
I know that $\frac{(m-1)!}{(m-n)!(n-1)!} + \frac{(m-1)!}{(m-n-1)!(n)!} = \frac{m!}{(n)!(m-n)!}$, but I am not sure on the intermediate steps. The only solution I am seeing involves finding a common ...
### How to prove $a^n < n!$ for all $n$ sufficiently large, and $n! \leq n^n$ for all $n$, by induction?
I have a couple things I want to prove. I'm pretty sure a proof by induction is the best route for these. First, I need to show that $5^n < n!$ from some $n_{0} > 0$. I'm choosing \$n_{0} = ...