# Tagged Questions

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

@Chris's_sis gave me following hint in a problem : $\frac{1}{ \displaystyle \binom{ p+k}{p}}- \frac{1}{ \displaystyle\binom{p+k+1}{p}} =\frac{ p}{p+1}\frac{ 1}{\displaystyle\binom{p-k-1}{p-1}}$ ...
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### 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 ...
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### 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 ...
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### 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?
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### 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. ...
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### 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 ...
How to prove these indentities? $\displaystyle \sum \limits_{k\geq0} {2n\choose 2k-1}{k-1\choose m-1}=2^{2n-2m+1}{2n-m\choose m-1}$ $\displaystyle \sum \limits_{k=0}^{m} {m\choose k}{n+k\choose ... 1answer 46 views ### Calculate sum wtih binomial coefficients I need help with finding the sum of$\sum \limits_{k=0}^{n} \frac{1}{k+1}{n\choose k}x^{k+1}$1answer 57 views ### Using combinatorial reasoning to show$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 ... 7answers 145 views ### Calculating$\binom{1}{2}$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 ... 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, $$\lim_{n\to\infty}\frac{1}{2n}\left(\log\left(2n!\right) - ... 1answer 80 views ### Falling factorial summands Representing the summand as a falling factorial Compute the sum$$\sum_{k=1}^n\frac{1}{(k+1)(k+2)}$$0answers 40 views ### Approximating the probability that a range bounds a given number, with very large numbers Let m numbers be chosen uniformly from 0,\dots,n-1 without replacement and then sorted in ascending order as \ell_0,\dots,\ell_{m-1}. Let there be b,e,x such that 0 \le b \le e \le m and 0 ... 3answers 159 views ### How does \tbinom{4n}{2n} relate to \tbinom{2n}{n}? I got this question in my mind when I was working on a solution to factorial recurrence and came up with this recurrence relation:$$(2n)!=\binom{2n}{n}(n!)^2$$which made me wonder: is there also a ... 3answers 110 views ### N women and N men. Groups of pairs. So we have N men and N women. We are creating groups of pairs. It is not necessary to use every man and woman. How many groups can we make ? So if we number them from 1 to N - let W_{1} be ... 1answer 296 views ### Purely combinatorial proof and simplification of identity involving factorials and summations While trying to decompose factorials into summations, I came up with the following identity$$(n+2)! = 2^{n+1} + \sum\limits_{k=0}^{n-1}\sum\limits_{i=0}^{n-1-k}\sum\limits_{S \subseteq ... 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 ... 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, $$(a)_n = a(a + 1)(a + 2) \cdots (a ... 3answers 78 views ### Is There a Way to Specify Limits On a Factorial If I want to be able to express a factorial -- let's say "20!" -- but with upper and lower limits such that the factorial is evaluated from Upper Limit, n1=20, through a Lower Limit, n2=10, for ... 1answer 25 views ### Number of orders and combinations I have just done these two questions and I have answers for them but I am not sure if they are correct. A jazz band is to give one concert in each of nine selected cities. Calculate the total ... 2answers 576 views ### How many perfect squares divide 1!2!3!4!5!6!7!8!9! What I naturally did was to find the prime factorisation of the product of factorials which is 2^{30}3^{13}5^5 7^3 . Clearly there is 15 unique perfect squares that divide 2^{30}, 6 unique ... 4answers 73 views ### Binomial expansions and factorials How to calculate$$\frac{n!}{n_1! n_2! n_3!}$$where n= n_1+n_2+n_3 for higher numbers n_1,n_2,n_3 \ge 100? This problem raised while calculating the possible number of permutations to a given ... 1answer 30 views ### Language to describe a number smaller than, but related to Bell number I understand that the Bell number B_n is the number of partitions of a set of size n. Despite my incredible ineptitude at combinatorics, I also understand most of how the binomial coefficient ... 0answers 70 views ### Find all possible solutions! Find solutions for$$^nP_r=s!$$For (n,r,s)\in \mathbb{N} I could find some trivial solutions (6,3,5)~,~(1,1,1) etc. 2answers 57 views ### Logic of statement I can see the mathematical implication but could not get the logic, why 5! is equal to ^6P_3? Please help proving why both the expressions are equal without mathematical manipulation!In any case, ... 0answers 40 views ### Combinatorics - find n! using inclusion-exclusion [duplicate] difficult question I need help with. We are asked to show that n! = \sum_{k=0}^n (-1)^k\binom{n}{k}(n-k)^n There is also a hint "try to think of the number of permutations of n elements using ... 1answer 109 views ### Euler numbers grow 2\left(\frac{2}{ \pi }\right)^{2 n+1}-times slower than the factorial? Stirling's approximation of the factorial for even numbers is given by$$ (2n)! \sim \left(\frac{2n}{e}\right)^{2n}\sqrt{4 \pi n}. \tag{1} $$Further, the Euler numbers grow quite rapidly for large ... 3answers 136 views ### Solve Algebraical.ly 0.5=\dfrac{365!}{365^{n}(365-n)!} How does one go about solving this equation? Not sure how to approach this as no factorials will cancel out. Im sorry I meant \dfrac{365!}{365^{n}(365-n)!}=0.5. 3answers 103 views ### Use a factorial argument to show that C(2n,n+1)+C(2n,n)=\frac{1}{2}C(2n+2,n+1) I need some help, showing that the left hand side is equivalent to the right hand side. I tried but I get stuck, I am not sure if I am on the right path. Here is my attempt: C(2n,n+1) + C(2n,n) ... 1answer 46 views ### Solving a permutation/Combination equation please help check if this would the correct way to solve this: ^nP_2 = ^{n+1}C_3. I want to solve for n. theoretically, I was thinking that: ^nP_k = k!\times ^nC_k hence: ... 1answer 122 views ### Asymptotics for sums involving factorials This question is rather general, but I have recently encountered the following situation in a variety of different settings. Let us suppose that we are given a complicated sum involving factorials ... 1answer 81 views ### Double factorial identity Does anyone know a strategy for proving$$ 2\cdot(2k-3)!!=\sum_{i=1}^{k-1}(2i-3)!!(2(k-i)-3)!!\binom{k}{i} $$for k\geq 2? Note that (-1)!!=1. Hints would be most appreciated. Full solutions not ... 4answers 188 views ### Is n \choose k defined when k < 0? What about n < k? I know that {n \choose 0} = 1, and this makes sense to me based on my understanding of combinatorics. But what about {n \choose -1}? My instinct is that this is undefined, since it is equivalent ... 1answer 47 views ### Counting permutations, with additional restrictions There are 10 slots and some marbles: 5 red, 3 blue, 2 green, how many ways can you fit those marbles into those slots? Those marbles fit in 10!/(5! 3! 2!) ways ... 2answers 204 views ### Simple Combinatorics Problem I've 'indirectly' studied combinatorics earlier in probability courses, but now it's part of the math course I'm taking. I always thought it was very hard, and well, here I am again... The problem ... 1answer 356 views ### Solving equation involving factorial I have the following problem:$$ N^n(N-n)!=A $$Where N and A are constants. I want to solve this equation for n (for a variation of the birthday problem), but I have little experience with ... 0answers 94 views ### Compute (-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!} Compute (-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!} Define a_{k,m}=\frac{(-1)^{k+m}(n+k-1)!(n+m-1)!}{(k+m-1)[(k-1)!(m-1)!]^2(n-m)!(n-k)!} Compute ... 3answers 199 views ### How Many Ways to Build a 6-Pack There is a beverage company here that claims to have a selection of 200 different beers. They have a special deal where you can build your own six pack at a discount. They advertise that there are ... 3answers 119 views ### Factorial Equality Problem I'm stuck on this problem, any help would be appreciated. Find all n \in \mathbb{Z} which satisfy the following equation:$${12 \choose n} = \binom{12}{n-2}$$I have tried to put each of them ... 2answers 65 views ### Identity of binomial series with factorial. I'm looking for a simple identity for the formula:$$ \sum_{k = 0}^{p} \binom{p}{k} \cdot k! \cdot x^k $$In words, I have p "players" who can choose to play or not (every player is represented by ... 2answers 68 views ### How to define a factorial using multiple sets I am currently studying a photography course, and I have run into a bit of difficulty with one of my projects in relation to combinatorics. I have a key rack and there are 39 hooks on this key rack. I ... 3answers 197 views ### Proving that \frac{(k!)!}{k!^{(k-1)!}} is an integer I have to prove that:$$\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}}$$But I think I just made it harder, and ... 3answers 281 views ### Compact formula for \sum_k k! [duplicate] Is there any compact formula for:$$\sum_{k=0}^n k!$$I've tried to find it using one method for summation, but I was able to receive only compact formula for \sum_k k! \cdot k = (n+1)!-1 I've ... 4answers 304 views ### Combinatorial proof to n! = (n-1)[(n-1)! + (n-2)!] It is for sure true that n! = (n-1)[(n-1)! + (n-2)!] Since: (n-1)(n-1)! + (n-1)(n-2)! = (n-1)(n-1)! + (n-1)! = (n-1)!(n-1+1) = (n-1)!n = n! Today my friend told me that there is a ... 1answer 267 views ### Handling summations with two variables If I have a summation with let's say x=0 \dots 500 and y=0\dots1500 500 \choose x 1500 \choose y \dfrac{1}{2^{500}}\dfrac{2^{1500-y}}{3^{1500}}, How would I handle the constant? If I ... 1answer 135 views ### Can this product be written so that symmetry is manifest? Let i, j, k be nonnegative integers such that i+j+k is even. The expression$$(-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}$$apparently computes the ... 0answers 40 views ### Bound on Permutations [duplicate] I am trying to prove the following inequality,$$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)} = ...
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 ...