# Tagged Questions

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### Proof of Pascal' identity

The identity $$\binom{x+1}{k}-\binom{x}{k}=\binom{x}{k-1}$$ is claimed to hold (using the binomial polynomials, considered as lying in $\mathbf{Q}[x]$) for $k$ at least $1$. Proof: by the usual ...
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### 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 ...
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### Prove the following : [duplicate]

Prove the following : $${{n}\choose{7}}-\left \lfloor{\frac{n}{7}}\right \rfloor$$ is divisible by 7.
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$$\left[\binom n{k-1} + \binom nk\right] + \left[\binom nk + \binom n{k+1}\right] = \binom{n+1}k + \binom{n+1}{k+1}$$ Can anyone else explain to me, without using Pascal's triangle, how this ...
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### 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 ...
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### Calculate sum wtih binomial coefficients

I need help with finding the sum of $\sum \limits_{k=0}^{n} \frac{1}{k+1}{n\choose k}x^{k+1}$
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### How to calculate this sum

How do you calculate this sum $\sum \limits_{k=1}^{n} \frac{k}{n^k}{n\choose k}$ ?
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### 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 $a^{n - k}b^k$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) = $\frac{2}{4} \times \frac{1}{3}$ = ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### “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 ...
### 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 ...
### 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 ...