# Tagged Questions

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 ...
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 ...
1k views

### Mathematical Induction Factorials, sum r(r!) =(n+1)! -1 [duplicate]

How do I prove that $$\sum\limits_{r=1}^{n} r(r!) = (n+1)!-1$$ I was able to get to factor: $LHS = k(k!) + (k+1)(k+1)!$ $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\, RHS = (k+2)! -1$
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### A Better Way to Solve this Factorial Problem?

I had a problem that asked me to find which of the following is larger: ${2013 \choose 500}$ or ${2013 \choose 1500}$ Beneath is my proof. I think it is correct (though your verification and ...
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### Proof by induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$

Prove via induction that$\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$ Having a very difficult time with this proof, have done pages of work but I keep ending up with 1/(k+2). Not sure when to ...
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### Prove that $\frac{n!}{(n-k)!} = n^{\underline{k}}$

I'm having some trouble proving the relation $$\frac{n!}{(n-k)!} = n^{\underline{k}}$$ Do you have to get into using gamma functions in order to prove this rigorously? Also, wikipedia seems to ...
344 views

### Evaluate $\displaystyle\sum_{k=1}^nk\cdot k!$

I discovered that the summation $\displaystyle\sum_{k=1}^n k\cdot k!$ equals $(n+1)!-1$. But I want a proof. Could anyone give me one please? Don't worry if it uses very advanced math, I can just ...
294 views

### Prove that $(a+1)(a+2)…(a+b)$ is divisible by $b!$ [duplicate]

The problem is following, prove that: $$(a+1)(a+2)...(a+b)\text{ is divisible by } b!\text{ for every positive integer a,b}$$ I've tried solving this problem using mathematical induction, but I ...
151 views

### Showing that $3n<n!$ whenever $n$ is an integer with $n \geq 7$

How can we show that: $$3n< n!$$ whenever $n$ is an integer such that $n \geq 7$ ? I was thinking that we can prove this by showing that such case is true with any integer above 7, but ...
155 views

### Prove that $n! ≥ (⌈n/2⌉)^{⌈n/2⌉}$ [closed]

Prove that : $n! ≥ (⌈n/2⌉)^{⌈n/2⌉}$
### Constructing $\mathbb N$ from the set of factorials
Let S be the set $\{0!, 1!, 2!, \ldots\}$. Is it possible to construct any positive integer using only addition, subtraction and multiplication, and using any element in S at most once? For example: ...