1
vote
1answer
63 views

Prove $1! + 2! + . . . + n! < (n + 1)!$ using mathematical induction [duplicate]

$1! + 2! + . . . + n! < (n + 1)!$ This question has left me stumped for quite some time. I am not sure how to approach it. (I am really bad at induction).
5
votes
1answer
75 views

Expressing a recursively defined function in terms of factorials or gamma function

Given the recursion $$f(n) = nf(n-1) + (n-1)f(n-2) $$ $$f(0) = 1, f(1) = 1$$ How exactly does one express the target function? I know that $$f(n) = nf(n-1)$$ gives rise to $$f(n) = ...
0
votes
1answer
36 views

Maximum value of function involving factorials

Define $$g_{(k,j)} = \frac{a^{n-k}b^k(k+n)!x^{k+n-j}}{k!(n-k)!(k+n-j)!}$$, where $n,k,j \in \Bbb{N}$ are fixed such that $(0 \leq x \leq a/b ),(b<a),(0 \leq k \leq n ),(2 \leq j \leq 2n),(0 \leq ...
0
votes
1answer
70 views

Find the largest integer $n$ such that $10^n$ divides $10^6!$

Let $N=10^6!$ Find the largest integer $n$ such that $10^n$ divides $N$. Furthermore, compute the first digit and the last non-zero digit of $N$. I have some ideas that you should be able to use ...
2
votes
0answers
58 views

How to prove these indentities? [closed]

How to prove these indentities? $\displaystyle \sum \limits_{k\geq0} {2n\choose 2k-1}{k-1\choose m-1}=2^{2n-2m+1}{2n-m\choose m-1}$ $\displaystyle \sum \limits_{k=0}^{m} {m\choose k}{n+k\choose ...
2
votes
1answer
50 views

Calculate sum wtih binomial coefficients

I need help with finding the sum of $\sum \limits_{k=0}^{n} \frac{1}{k+1}{n\choose k}x^{k+1}$
0
votes
2answers
45 views

How to calculate this sum

How do you calculate this sum $ \sum \limits_{k=1}^{n} \frac{k}{n^k}{n\choose k}$ ?
0
votes
7answers
152 views

Calculating $\displaystyle\sum_{i=1}^{n} \binom{i}{2}$

Show $\displaystyle\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$. I'm thinking right now (though not getting anywhere with it) that I want to expand out the summation portion to $i!/2!(i-2)!$ and ...
1
vote
1answer
26 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 ...
0
votes
1answer
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 ...
2
votes
2answers
56 views

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 ...
1
vote
1answer
110 views

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 ...
5
votes
2answers
112 views

Prove $((n+1)!)^n < 2!\cdot4!\cdots(2n)!$

so I know I need to prove this via induction, but I am somewhat stuck. Here is what I have does so far. Let $p(n) = (n+1)!^n \le 2!\cdot4!\cdot\ldots\cdot(2n)!$ $p(2) = 3!^2\le 2!\cdot4!$ Assume ...
2
votes
3answers
2k views

Sum of reciprocals of factorials

Could you help me count this sum: $$ \sum_{n=1}^{9} \frac{1}{n!} $$ I don't think I can use binomial coefficients.
2
votes
3answers
246 views

Algebraic manipulation of binomial theorem

Prove, by algebraic manipulation, that: \[ {{2n} \choose {n}} + {{2n} \choose {n+1}}={1\over2} {{2n+2} \choose {n+1}} \]
11
votes
1answer
631 views

Factorial canceling on expansion of binomial coefficients on Concrete Mathematics

On Concrete Mathematics section 5.5, which is teaching the hypergeometric functions, generalized factorials is defined as: \[ \frac 1 {z!} = \lim_{n \to \infty} \binom{n+z}{n}n^{-z} \] where \[ ...
0
votes
2answers
129 views

What is the minimum number of friends Sally can have if she can invite a different group of friends to her house every night for a year?

The question: Sally has $N$ friends and likes to invite them over in small groups for dinner. She calculated that she can invite a different group of $3$ friends to dinner at her house every night for ...
3
votes
0answers
195 views

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 ...
2
votes
1answer
382 views

Use of algebra and factorials for a question related to proof by induction

$$ \begin{align*} &= (n+1)! − 1 + ( (n+1) · (n+1)! )\\ &= (n+1)! (1+n+1) − 1\\ &= (n+1)! (n+2) − 1\\ &= (n+2)! − 1\\ \end{align*} $$ I'm confused at how the first ...