1
vote
2answers
20 views

Counting Problem using Permutations

The Question was: In how many ways can the letters of the English alphabet be arranged so that there are exactly 10 letters between a and z? My approach was the following: In between a and z, there ...
3
votes
4answers
111 views

Proving Combinatorical Summation: $n!=\sum_{k=0}^n(-1)^k\binom{n}{k}(n-k)^n$ [duplicate]

been stuck with this question for the last few hours, any help would be appreciated. $$ {\large n! = \sum_{k = 0}^{n}\left(-1\right)^{k}{\,n\, \choose \,k\,} \left(\,n - k\,\right)^{n}} $$ what I ...
2
votes
2answers
63 views

What is $\binom{a}b$ with $a<b$?

@Chris's_sis gave me following hint in a problem : $\frac{1}{ \displaystyle \binom{ p+k}{p}}- \frac{1}{ \displaystyle\binom{p+k+1}{p}} =\frac{ p}{p+1}\frac{ 1}{\displaystyle\binom{p-k-1}{p-1}}$ ...
2
votes
0answers
38 views

Find k such that, $\displaystyle\frac{n^{\underline k}}{n^k}<a$

How to find k here $\displaystyle\frac{n^{\underline k}}{n^k}<a$ ? With, $n^{\underline k}=n\cdot(n-1)\cdot...(n-k+1)$ and $(n>k,\ a>0)$ Of course if $k\approx n/2$ the inequality ...
8
votes
1answer
142 views

One-Line Proof for $n! \geq (\frac n e)^n$

I was told to find a one-line proof for $n! \geq (\frac n e)^n$. I'm advised that Stirling's formula is not helpful. I've spent a little bit of time on it, but the solution is not coming to me. I feel ...
6
votes
1answer
49 views

maximize a function which contains factorials

Suppose I have a function $$ f(k) = \binom{500}{k} \binom{500}{1100-3k}$$ where $k$ is an integer from $200$ to $366$. How can I find the maximum analytically?
1
vote
0answers
9 views

Simplifcation of the Function Required

I want to simplify the following expression $$ \sum_{x=0}^{D+m} \frac {x!} {(n+1)_x} (\frac {\theta_{eff}} {\mu})^x {D+m \choose x} $$ The parameters $D, m, n, \theta_{eff}, \mu, $ are constants. ...
-1
votes
3answers
115 views

How to interpret $(2n)!$

It's all in question: how to interpret the factorial from $2n$? Is $(2n)!$ equal to $n!\times n!$ ? The problem is in Combinations if the combinations is $\binom{2n}3$. P.S. The main problem is ...
2
votes
0answers
55 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
46 views

Calculate sum wtih binomial coefficients

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

Using combinatorial reasoning to show $n!=\binom{n}{0}D_n+\binom{n}{1}D_{n-1}+\dots+\binom{n}{n}D_0$

How can one use combinatorial reasoning to show that $$n!=\dbinom{n}{0}D_n+\dbinom{n}{1}D_{n-1}+\dbinom{n}{2}D_{n-2}+....+\dbinom{n}{n-1}D_1+\dbinom{n}{n}D_0$$ Now $D$ stands for deranged which is a ...
0
votes
7answers
149 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 ...
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) - ...
0
votes
1answer
80 views

Falling factorial summands

Representing the summand as a falling factorial Compute the sum $$\sum_{k=1}^n\frac{1}{(k+1)(k+2)}$$
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 ...
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 ...
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
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
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 ...
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
1answer
25 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 ...
3
votes
2answers
577 views

How many perfect squares divide 1!2!3!4!5!6!7!8!9!

What I naturally did was to find the prime factorisation of the product of factorials which is $ 2^{30}3^{13}5^5 7^3 $. Clearly there is 15 unique perfect squares that divide $2^{30}$, 6 unique ...
2
votes
4answers
73 views

Binomial expansions and factorials

How to calculate $$\frac{n!}{n_1! n_2! n_3!}$$ where $n= n_1+n_2+n_3$ for higher numbers $n_1,n_2,n_3 \ge 100$? This problem raised while calculating the possible number of permutations to a given ...
0
votes
1answer
30 views

Language to describe a number smaller than, but related to Bell number

I understand that the Bell number $B_n$ is the number of partitions of a set of size $n$. Despite my incredible ineptitude at combinatorics, I also understand most of how the binomial coefficient ...
2
votes
0answers
70 views

Find all possible solutions!

Find solutions for $$^nP_r=s!$$ For $(n,r,s)\in \mathbb{N}$ I could find some trivial solutions $(6,3,5)~,~(1,1,1)$ etc.
2
votes
2answers
57 views

Logic of statement

I can see the mathematical implication but could not get the logic, why $5!$ is equal to $^6P_3$? Please help proving why both the expressions are equal without mathematical manipulation!In any case, ...
2
votes
0answers
40 views

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 ...
3
votes
1answer
109 views

Euler numbers grow $2\left(\frac{2}{ \pi }\right)^{2 n+1}$-times slower than the factorial?

Stirling's approximation of the factorial for even numbers is given by $$ (2n)! \sim \left(\frac{2n}{e}\right)^{2n}\sqrt{4 \pi n}. \tag{1} $$ Further, the Euler numbers grow quite rapidly for large ...
1
vote
3answers
136 views

Solve Algebraical.ly $0.5=\dfrac{365!}{365^{n}(365-n)!} $

How does one go about solving this equation? Not sure how to approach this as no factorials will cancel out. Im sorry I meant $\dfrac{365!}{365^{n}(365-n)!}=0.5$.
3
votes
3answers
103 views

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: $C(2n,n+1) + C(2n,n)$ ...
2
votes
1answer
46 views

Solving a permutation/Combination equation

please help check if this would the correct way to solve this: $^nP_2 = ^{n+1}C_3$. I want to solve for $n$. theoretically, I was thinking that: $^nP_k = k!\times ^nC_k $ hence: ...
3
votes
1answer
123 views

Asymptotics for sums involving factorials

This question is rather general, but I have recently encountered the following situation in a variety of different settings. Let us suppose that we are given a complicated sum involving factorials ...
0
votes
1answer
81 views

Double factorial identity

Does anyone know a strategy for proving $$ 2\cdot(2k-3)!!=\sum_{i=1}^{k-1}(2i-3)!!(2(k-i)-3)!!\binom{k}{i} $$ for $k\geq 2$? Note that $(-1)!!=1$. Hints would be most appreciated. Full solutions not ...
2
votes
4answers
190 views

Is $n \choose k$ defined when $k < 0$? What about $n < k$?

I know that ${n \choose 0} = 1$, and this makes sense to me based on my understanding of combinatorics. But what about ${n \choose -1}$? My instinct is that this is undefined, since it is equivalent ...
2
votes
1answer
49 views

Counting permutations, with additional restrictions

There are 10 slots and some marbles: 5 red, 3 blue, 2 green, how many ways can you fit those marbles into those slots? Those marbles fit in 10!/(5! 3! 2!) ways ...
2
votes
2answers
206 views

Simple Combinatorics Problem

I've 'indirectly' studied combinatorics earlier in probability courses, but now it's part of the math course I'm taking. I always thought it was very hard, and well, here I am again... The problem ...
6
votes
1answer
359 views

Solving equation involving factorial

I have the following problem: $$ N^n(N-n)!=A $$ Where $N$ and $A$ are constants. I want to solve this equation for $n$ (for a variation of the birthday problem), but I have little experience with ...
2
votes
0answers
94 views

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$ Define $a_{k,m}=\frac{(-1)^{k+m}(n+k-1)!(n+m-1)!}{(k+m-1)[(k-1)!(m-1)!]^2(n-m)!(n-k)!}$ Compute ...
5
votes
3answers
199 views

How Many Ways to Build a 6-Pack

There is a beverage company here that claims to have a selection of 200 different beers. They have a special deal where you can build your own six pack at a discount. They advertise that there are ...
2
votes
3answers
119 views

Factorial Equality Problem

I'm stuck on this problem, any help would be appreciated. Find all $n \in \mathbb{Z}$ which satisfy the following equation: $${12 \choose n} = \binom{12}{n-2}$$ I have tried to put each of them ...
3
votes
2answers
65 views

Identity of binomial series with factorial.

I'm looking for a simple identity for the formula: $$ \sum_{k = 0}^{p} \binom{p}{k} \cdot k! \cdot x^k $$ In words, I have $p$ "players" who can choose to play or not (every player is represented by ...
3
votes
2answers
68 views

How to define a factorial using multiple sets

I am currently studying a photography course, and I have run into a bit of difficulty with one of my projects in relation to combinatorics. I have a key rack and there are 39 hooks on this key rack. I ...
15
votes
3answers
197 views

Proving that $\frac{(k!)!}{k!^{(k-1)!}}$ is an integer

I have to prove that: $$\frac{(k!)!}{k!^{(k-1)!}} \in \Bbb Z$$ for any $k \geq 1, k \in \Bbb N$ Tried doing $t = k!$ which would give $$\frac{t!}{t^{t/k}}$$ But I think I just made it harder, and ...
7
votes
3answers
281 views

Compact formula for $\sum_k k!$ [duplicate]

Is there any compact formula for: $$\sum_{k=0}^n k!$$ I've tried to find it using one method for summation, but I was able to receive only compact formula for $\sum_k k! \cdot k = (n+1)!-1$ I've ...
7
votes
4answers
304 views

Combinatorial proof to $n! = (n-1)[(n-1)! + (n-2)!]$

It is for sure true that $n! = (n-1)[(n-1)! + (n-2)!]$ Since: $(n-1)(n-1)! + (n-1)(n-2)! = $ $(n-1)(n-1)! + (n-1)! =$ $ (n-1)!(n-1+1) = (n-1)!n = n! $ Today my friend told me that there is a ...
0
votes
1answer
268 views

Handling summations with two variables

If I have a summation with let's say $x=0 \dots 500$ and $y=0\dots1500$ $500 \choose x$ $ 1500 \choose y$ $\dfrac{1}{2^{500}}\dfrac{2^{1500-y}}{3^{1500}}$, How would I handle the constant? If I ...
10
votes
1answer
135 views

Can this product be written so that symmetry is manifest?

Let $i,$ $j,$ $k$ be nonnegative integers such that $i+j+k$ is even. The expression $$(-1)^{j+k}\binom{i+j+k}{i,j,k}\prod_{\ell=0}^{k-1} \frac{i-j+k-2\ell-1}{i+j+k-2\ell-1}$$ apparently computes the ...
3
votes
0answers
40 views

Bound on Permutations [duplicate]

I am trying to prove the following inequality, $$n^{(l)} = n(n-1)\cdots(n-l+1) \geq \frac{n^l}{e}\quad\text{ for }\quad 2 \leq l \leq \sqrt{n}\;.$$ So my approach is to observe that $n^{(l)} = ...
2
votes
4answers
3k views

Solving Equations with Factorials

I am attempting to solve an equation ${{n-2} \choose {2}} + {{n-3} \choose {2}} + {{n-4} \choose {2}} = 136$. With the formula for a combination being $\frac{n!}{r!(n - r)!}$, I simplified the given ...