0
votes
2answers
70 views

Show that $\binom{n}{k}< \binom{n}{k+1}$ if and only if $k < (n-1)/2$ [closed]

Show that $\binom{n}{k} < \binom{n}{k+1}$ if and only if $k < \frac{n-1}{2}$ and then use this to deduce that the maximum of $\binom{n}{k}$ for $k=0,1,\dots,n$ is $\binom{n}{\lfloor ...
2
votes
1answer
79 views

A generalized combinatorial identity for a sum of products of binomial coefficients

I have the following question. For given natural numbers $n$ and $d$, let $a_1,a_2,..., a_r$ be fixed integers such that $a_1+\cdots+a_r=d$. Let $A=\{(i_1,..,i_r)~|~0\le i_j\le n~ \text{and}~ ...
2
votes
0answers
32 views

Inequality in matroid theory

Working on a proof in matroid theory I found there is a smooth map from an open set of $(\mathbb{C}^{\ast})^{(d−1)(n−d−1)}$ to a disjoint union of tori $(S^{1})^{\binom{n}{d}-n}.$ As a direct ...
1
vote
1answer
32 views

Equality of binomial coefficients

I have seen that the following equations are equal, but are wondering how this is shown ${n \choose m} \cdot 1 \cdot 3 \cdots (2m-1)\cdot 1 \cdot 3 \cdots (2n-2m-1) = \frac{n!}{2^n} {2m \choose m} ...
9
votes
3answers
303 views

Help with combinatorial proof of identity: $\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k} \binom{n}{k} = \sum_{k=1}^{n} \frac{1}{k}$

How to prove this identity? Can someone please give me some insight ? $$\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k} \binom{n}{k} = \sum_{k=1}^{n} \frac{1}{k}$$
0
votes
2answers
60 views

How many coefficients are in the expansion $(x + y + z)^{10}$

I need to find the number of coefficients in the expansion $(x + y + z)^{10}$. I had this exercise on a recent assignment. The answer I gave is: $3^{10} = \binom {3 + 10 - 1}{10} = \binom{12}{10} = ...
2
votes
0answers
55 views

Prove the following : [duplicate]

Prove the following : $$ {{n}\choose{7}}-\left \lfloor{\frac{n}{7}}\right \rfloor $$ is divisible by 7.
1
vote
1answer
45 views

Addition of Binomial Coefficients

$$\left[\binom n{k-1} + \binom nk\right] + \left[\binom nk + \binom n{k+1}\right] = \binom{n+1}k + \binom{n+1}{k+1}$$ Can anyone else explain to me, without using Pascal's triangle, how this ...
4
votes
4answers
114 views

Combinatorial explanation for why $n^2 = {n \choose 2} + {n+1 \choose 2}$

An exercise in the first chapter of Discrete Mathematics, Elementary and Beyond asks for a proof of the following identity: $$ {n \choose 2} + {n+1 \choose 2} = n^2 $$ The algebraic solution is ...
0
votes
0answers
22 views

closed form of a specific crazy summation?

How can I find the closed form of $f_2 + f_4 + ...+ f_{2m}$ where $\sum\limits_{m=1}^\infty f_{2m} = u_{2m-2}- u_{2m} $ where $u_{2m} = \binom{2m}{m} 2^{-(2m)}$ and $u_{2m-2} = \binom{2m-2}{m-1} ...
2
votes
0answers
56 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
47 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
44 views

How to calculate this sum

How do you calculate this sum $ \sum \limits_{k=1}^{n} \frac{k}{n^k}{n\choose k}$ ?
2
votes
2answers
61 views

find the coefficient of the given term when the expression is expanded by the binomial theorem

I am just trying to understand why the term is $\binom{15}8$(3p$^2$ - 2q)$^7$. I need to find the coefficient in $p^{16}q^7$ in $(3p^2 - 2q)^{15}$ So, I know that $n = 15$ and I have $a^{n - k}b^k$ ...
1
vote
1answer
31 views

I want to prove this identity involving the binomial coefficients

Can you help me prove the following identity? I know it holds because I simulated it. For positive integers $n,m,k$ and for $i=0,\ldots,n$ and for $n \leq m$ we have: $$\sum_{j=0}^i (-1)^{i+j}\binom ...
0
votes
7answers
151 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 ...
3
votes
0answers
39 views

How to show that $\sum\limits_{k=0}^{\lfloor0.999n\rfloor}\binom{2n}{k} < \binom{2n}{n} $ holds for large n

It seems logical to me since $\binom{2n}{n}$ is in the middle of the row in pascal triangle; therefore, the largest, and for large n the sum adds only the small ones on the left. But I do not have any ...
2
votes
1answer
197 views

How to solve this recurrence relation with Sigma notation (f(n, m) = f(n - 1, m) + f(n, m- 1) + c?

This recurrence relation was inferred from the function $f(n, m) = f(n - 1, m) + f(n, m-1) + c$. After expanding the latter, I ended up with the following: $$f(n,m)=\begin{cases} 0,&\text{if ...
3
votes
3answers
264 views

What is the coefficient of the term $x^4 y^5$ in $(x+y+2)^{12}$?

What is the coefficient of the term $x^4 y^5$ in $(x+y+2)^{12}$? How can we calculate this expression ? I've applied the binomial theorem formula and got $91$ terms but I am not sure if it is right ...
2
votes
2answers
113 views

Infinite Sum with Combination

I am trying to figure out what the following sum converges to: $$\sum_{n=0}^\infty {6+n\choose n}x^n(6+n),\qquad\qquad0<x<1$$ An answer would be great, but if you have an explanation, that'd ...
1
vote
4answers
312 views

Calculate sum $\sum\limits_{k=0}k^2{{n}\choose{k}}3^{2k}$.

I need to find calculate the sum Calculate sum $\sum\limits_{k=0}k^2{{n}\choose{k}}3^{2k}$. Simple algebra lead to this ...
2
votes
1answer
73 views

Finding the super-mean (NOT the mean) of a set of numbers.

the super-mean is found by grouping pairs of numbers and finding the average successively until there is just one number. For example, $$(1-2-3-4-5) \to ((1+2)/2,(2+3)/2,(3+4)/2,(4+5)/2) \\ ...
1
vote
1answer
31 views

Largest K-multiple free set out of a fully ordered set

i'm struggling conceptually with this problem, i don´t know how to approach it in a clever way (without a computer, or at least without a brilliant algorithm). Mathematicians defined a k-multiple set ...
3
votes
1answer
100 views

Proof sought for a sum involving binomials that simplifies to 1/2

A proof of: $$\begin{align*}(1/2)^{2m+1} \sum_{k=0}^{m} \binom{m}{k} \sum_{j=0}^{k} \binom{m+1}{j} = \frac{1}{2} \end{align*} $$ Conjecture based on the following Maple code: ...
2
votes
1answer
130 views

Identify $\sum(-1)^{n-k}2^{2k}\binom{n+k}{2k}$

Does anybody know what the following sum evaluates to? $$ \sum_{k=0}^n{(-1)^{n-k}}2^{2k}\binom{n+k}{2k} $$
1
vote
1answer
77 views

Show binomial coefficient for $x^n$ in the expansion $(1+x)^r(1+x)^s$

Show that the binomial coefficient for $x^n$ in the expansion $(1+x)^r(1+x)^s$ is $\sum_{k=0}^{n}\binom{r}{k}\binom{s}{n-k}$. I dont know how to reach that. What i see is that: ...
3
votes
1answer
190 views

Find the sum of the series.

I need to find the following sum: $$\sum_{s=0}^{n+1}{(-1)}^{n-s}4^s\binom{n+s+1}{2s}$$ First I tried to simplify this: $$\begin{split} \sum_{s=0}^{n+1}{(-1)}^{n-s}4^s\binom{n+s+1}{2s} &= ...
7
votes
1answer
128 views

A finite sum involving the binomial coefficients and the harmonic numbers

Wikipedia has a proof of the identity $$ H_{n} =\sum_{k=1}^{n} (-1)^{k-1} \binom{n}{k} \frac{1}{k}$$ http://en.wikipedia.org/wiki/Harmonic_number#Calculation Curiously, there is also the identity ...
0
votes
3answers
117 views

Probability of choosing coins from a bag (why doesn't binomial coefficient work?)

Studying for my exam and would appreciate some help. I have a bag with 2 pennies, 1 nickel and 1 dime. I pick 2 at random. The solutions say: Pr(PP) = $\frac{2}{4} \times \frac{1}{3}$ = ...
0
votes
2answers
70 views

generating function and binomial distribution - counting

I am trying to understand generating function. I have the following problem: There are 50 students in the International Mathematical Olympiad (IMO) training programme. 6 of them are to be selected to ...
0
votes
2answers
31 views

negative binomial distribution problem

Find the probability that you find 2 defective tires before 4 good ones. There is a chance of a tire being defective at a rate of 5%. From my understanding with the negative binomial distribution we ...
2
votes
2answers
40 views

Expansion Coefficient needed

This is probably something very easy, but wth... my mind is totally stuck right now. I need to find the coefficient of $x^{11}$ of the expansion $(x^2 + 2\frac yx)^{10}$ Well I know that the answer ...
3
votes
1answer
110 views

Pascal Triangle general formula

I'm working on a presentation on the Binomial Theorem for my Algebra 2 class and while writing Pascal's Triangle, I came across one of the properties that I haven't seen in a while. That being ...
0
votes
0answers
95 views

Combinatorial word problems (Discrete math)

I have a problem with writing the word problems to which the answers are the following expressions. I am not sure if these answers sound right. I am not good with writing questions to these ...
1
vote
2answers
93 views

Combinatorial proof with binomial coefficients

I need to prove this with combinatorial arguments. I don't know how to start. $$ \sum_{j = r}^{n + r - k}{j - 1 \choose r - 1}{n - j \choose k - r} = {n \choose k}\,, \qquad\qquad 1\ \leq\ r\ \leq\ ...
1
vote
1answer
55 views

Why is $ 2\binom nm^2<n^{2m}$?

$\forall n\geq2 \forall m\geq2,$ $$ 2\binom nm^2<n^{2m}.$$ Why is the above inequality, which is equivalent to $ \binom nm<\frac{n^m}{\sqrt 2}$, true?
1
vote
3answers
73 views

Why is this binomial coefficient bounded thus?

Source: Miklos Bona, A Walk Through Combinatorics. $$ \forall k\geq 2,\binom{2k-2}{k-1}\leq4^{k-1}.$$ The RHS is the upper bound of the Ramsey number $R(k,k)$. How can I prove the inequality ...
1
vote
1answer
709 views

binomial distribution(overbooking plane tickets)

I am having trouble with binomial distribution and this problem: an airplane has 200 seats, but 202 tickets are sold. Assume passengers do not show up with a probability of .03 independently. What is ...
1
vote
0answers
54 views

Find asymptotics in a given form $n=(e+o(1))^{f(s)}$

Let $p\to\infty$, $s={\binom {p^4} p}$ and $n={\binom {p^4}{p^2}}$. Find a function $f(s)$ in the following form $$\large n=(e+o(1))^{f(s)}$$ I've tried to use the followinf asymptotics for ...
2
votes
4answers
273 views

How to determine the coefficient of binomial

Suppose I have $\left(x-2y+3z^{-1}\right)^4$ How to determine coefficient binom of $xyz^{-2}$? I've tried using trinominal expansion like this: $\displaystyle \frac{4!}{1!1!1!} (1)^1 (-2)^1 (3)^2$
3
votes
1answer
107 views

Find the constant $c$ in the equation $\max_{a\le n/2}\frac{C_n^a}{\sum_{k=0}^{\lfloor{a/3}\rfloor}C_n^k}=(c+o(1))^n.$

Find the constant $c$ in the equation $$\max_{a\le n/2}\frac{C_n^a}{\sum_{k=0}^{\lfloor{a/3}\rfloor}C_n^k}=(c+o(1))^n.$$ I've tried to use this asymptotics $$C_n^k \sim \frac{n^m}{m!} \sim e^{m\ln n ...
0
votes
1answer
133 views

Evaluating a sum with binomial coefficients

I have come across the following sum evoking the binomial theorem: $$\sum_{k=1}^n {n \choose k} \frac{1}{k^r} a^k b^{n-k},$$ where $r > 0$ is a positive real constant and $a,b \in \mathbb{R}$ are ...
0
votes
1answer
70 views

A few questions relating to counting for midterm practise exam?

I'm doing some questions for my midterm practise exam (multiple choice) for discrete structures and would appreciate some help (My answer is bolded): Using the 26-letter alphabet {a,b,c,...,z}, how ...
5
votes
4answers
233 views

Prove $\binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a}$

I want to prove this equation, $$ \binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a} $$ I thought of proving this equation by prove that you are using different ways to count the same set of ...
1
vote
2answers
1k views

Prove using the binomial theorem

I'm studying for a midterm and need some help with proving summation using the binomial theorem. $\sum\limits_{k=0}^n {n \choose k} 2^k = 3^n$ This is what I'm thinking so far: In the binomial ...
0
votes
3answers
128 views

Help with binomial coefficients using binomial theorem

I am studying for an upcoming test and I was having trouble with this practice problem: Determine the coefficient of $x^{111}y^{444}$ in the expansion of $(17x + 71y)^{555}$. I am thinking of using ...
3
votes
2answers
118 views

“Upper summation” binomial identity: different version from “Concrete Mathematics”

The book "Concrete Mathematics: A Foundation for Computer Science", 2nd Edition - authored by Ronald L. Graham, Donald E. Knuth, Oren Patashnik - has, in its page 174, a table called: "Table 174 The ...
1
vote
1answer
105 views

Recursive and closed form solution for choosing $n$ pairs/triplets.. of $kn$ elements.

I stumbled apon an interesting question: How many ways are there to arrenge $kn$ elements into $n$ sets, $k$ elements each? There should be a recursive and closed form solution for $g_k(n)$. For ...
1
vote
2answers
121 views

Prove combinatoric inequality: ${n \choose {j+k}}\le {n \choose j}{{n-j}\choose k}$

How can one prove the following combinatoric inequality? $${n \choose {j+k}}\le {n \choose j}{{n-j}\choose k}$$ My line of thought was: $n$ people applied for an interview for a company. (And the ...
1
vote
2answers
422 views

Algebraic proof of $\sum_{i=0}^k{{n \choose i}{m \choose {k-i}}}= {{m+n}\choose k}$

I can't figure out an algebraic proof for the following identity: $$\sum_{i=0}^k{{n \choose i}{m \choose {k-i}}}= {{m+n}\choose k}$$ Combinatorical solution: We can see that as choosing some from ...