# Tagged Questions

Coefficients involved in the Binomial Theorem. $\binom{n}{k}$ counts the subsets of size $k$ of a set of size $n$.

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### Evaluating Sums Algebraically or Combinatorially

Consider (1) $$\sum_{k=0}^{n}\binom{n}{k}2^{k-n}$$ (2) $$\sum_{k=0}^{n}\binom{n}{k}\frac{k!}{(n+k+1)!}$$ These sums appear too difficult (in my mind) to evaluate combinatorially. What are some ...
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### Why does $\sum\limits_{i=0}^k {k\choose i}=2^k$ [duplicate]

Possible Duplicate: Proving a special case of the binomial theorem Can anyone explain to me why $$\sum\limits_{i=0}^k {k\choose i}=2^k\,?$$ Thanks in advance
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### How do I show that $\sum_{k = 0}^n \binom nk^2 = \binom {2n}n$? [duplicate]

$$\sum_{k = 0}^n \binom nk^2 = \binom {2n}n$$ I know how to "prove" it by interpretation (using the definition of binomial coefficients), but how do I actually prove it?
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### What is the sum $\sum_{k=0}^{n}k^2\binom{n}{k}$? [duplicate]

What should be the strategy to find $$\sum_{k=0}^{n}k^2\binom{n}{k}$$ Can this be done by making a series of $x$ and integrating?
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### Prove by Combinatorial Argument that $\binom{n}{k}= \frac{n}{k} \binom{n-1}{k-1}$

This is a question from my first proofs homework and I am confused about the combinatorial argument aspect. I already did the algebraic proof. I think I am supposed to put into words what both sides ...
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### Property of a polynomial $f\in\mathbb{Q}[X]$ such that $f(n)\in\mathbb{Z}$ for all $n\in\mathbb{Z}$?

We can always view $\binom{x}{k}$ as a polynomial in $x$ of degree $k$. With this in mind, why is it so that a polynomial $f\in\mathbb{Q}[x]$ is such that $f(n)\in\mathbb{Z}$ for all $n\in\mathbb{Z}$ ...
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### Combinatorial proof of $\binom{n}{k} = \binom{n}{n-k}$

How do I prove this combinatorially? $$\displaystyle \binom{n}{k} = \binom{n}{n-k}$$
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### Find the sum of the $\sum_{m=k}^{+\infty}\binom{m}{k}(1-p)^k\cdot p^{m-k}$

Let $0<p<1$,Find the sum $$\sum_{m=k}^{+\infty}\binom{m}{k}(1-p)^k\cdot p^{m-k}$$
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### Curious Binomial Coefficient Identity

Consider the following set of identities: ${m+1\choose 1}={m\choose 1}+1$, ${m+1\choose 2}=2\binom m 2 - {m-1\choose 2}+1$, ${m+1\choose 3}=3\binom m3-3{m-1\choose 3}+{m-2\choose 3}+1$, ... This set ...
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### Solve for $n$, where $n$ is a positive integer

I have $${n \choose 2} = 21$$ and as the title mentions I have to solve for $n$, but so far all I have managed to get to is $$n^2 -n =42$$ and from there I'm completely lost. Any hints would ...
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### Binomial Summation

The sum $$1 + {n \choose 1}\cos \theta + {n \choose 2}\cos 2\theta + \cdots+ {n \choose n}\cos n\theta$$ is? I try to write this as the real part of $(1 + \cos \theta + i\sin \theta)^n$ but then I'...
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### Find a simple formula for

$$\binom{n}{0}\binom{n}{1}+\binom{n}{1}\binom{n}{2}+...+\binom{n}{n-1}\binom{n}{n}$$ All I could think of so far is to turn this expression into a sum. But that does not necessarily simplify the ...
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We must choose a 5-member team from 12 girls and 10 boys. How many ways are there to make the choice so that there are no more than 3 boys on the team? The correct answer is $\binom{22}{5} - \binom{... 2answers 343 views ### Combinatorial Proof for a$ p\mid\binom{p}{k} \ \ \ \ \ 0<k<p$. I'm looking for a combinatorial proof to the following statement: $$p\mid\binom{p}{k} \ \ \ , \ \ 0<k<p \ \ \ \ \ \ \text{and} \ \ p \ \text{is prime}.$$ Thank you. 2answers 1k views ### Factorial division using Pascal's triangle. I want to get values of factorial divisions such as 100!/(2!5!60!)(the numbers in the denominator will all be smaller than the numerator, and the sum of the ... 2answers 42 views ### Trouble understanding how this identity is derived:$\sum_{j=0}^{\infty}\binom{a+j}{j}x^j=(1-x)^{-a-1}$$$\sum_{j=0}^{\infty}\binom{a+j}{j}x^j=(1-x)^{-a-1}$$ The$-a-1$is throwing me off. Can anyone help me understand this identity. I have tried letting$m=-a-1$and then applying the binomial theorem,... 3answers 135 views ### Showing${n + a - 1 \choose a - 1} = \sum_{k = 0}^{\left\lfloor n/2 \right\rfloor} {a \choose n-2k}{k+a-1 \choose a-1}$Prove that for integers$n \geq 0$and$a \geq 1$, $${n + a - 1 \choose a - 1} = \sum_{k = 0}^{\left\lfloor n/2 \right\rfloor} {a \choose n-2k}{k+a-1 \choose a-1}.$$ I figured I'd post this question, ... 2answers 142 views ### What does$\lim \limits_{n\rightarrow \infty }\sum \limits_{k=0}^{n} {n \choose k}^{-1}$converge to (if it converges)? [duplicate] How we can show if the sum of $$\lim_{n\rightarrow \infty }\sum_{k=0}^{n} \frac{1}{{n \choose k}}$$ converges and then find the result of the sum if it converges? Thanks for any help. 4answers 142 views ### A binomial inequality with factorial fractions Prove that $$\left(1+\frac{1}{n}\right)^n<\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!}$$ for$n>1 , n \in \mathbb{N}$. 3answers 698 views ### No closed form for the partial sum of${n\choose k}$for$k \le K$? In Concrete Mathematics, the authors state that there is no closed form for $$\sum_{k\le K}{n\choose k}.$$ This is stated shortly after the statement of (5.17) in section 5.1 (2nd edition of the book).... 3answers 448 views ### Combinatorial proof for$\binom{n}{a}\binom{a}{k}\binom{n-a}{b-k} = \binom{n}{b}\binom{b}{k}\binom{n-b}{a-k}$? I have to prove the following using a combinatorial proof:$\binom{n}{a}\binom{a}{k}\binom{n-a}{b-k} = \binom{n}{b}\binom{b}{k}\binom{n-b}{a-k}$Ok, so here is what I have worked out so far: We ... 3answers 2k views ### Combination problem with constraints You have four containers and one pitcher of water that holds 100L. Each container has different capacities with maximums of, say...70L, 45L, 33L and 11L levels respectively. What is the formula that ... 2answers 6k views ### Number of even and odd subsets [duplicate] Suppose we have the following two identities:$\displaystyle \sum_{k=0}^{n} \binom{n}{k} = 2^n\displaystyle \sum_{k=0}^{n} (-1)^{k} \binom{n}{k} = 0$The first says that the number of subsets ... 2answers 39 views ### Let$f(n)=\sum_{r=0}^{n}\sum_{k=r}^{n}\binom{k}{r}$.If$f(n)=2047,$then find the value of$n.$Let$f(n)=\sum_{r=0}^{n}\sum_{k=r}^{n}\binom{k}{r}$.If$f(n)=2047,$then find the value of$n.f(n)=\sum_{r=0}^{n}\sum_{k=r}^{n}\binom{k}{r}=\sum_{k=0}^{n}\binom{k}{0}+\sum_{k=1}^{n}\binom{k}{1}+\...
I believe the following is an identity (I've tested with a few random $m$ and $n$ values, could be wrong though): $$\sum_{k= 0}^{\infty}{m \choose k}{n \choose k}k=n\binom{m+n-1}{m-1}$$ but I'm not ...