# 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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### Combinatorial proof of $\sum_{i=0}^k\binom{m+k-i-1}{k-i}\binom{n+i-1}{i}=\binom{m+n+k-1}{k}$?

Please provide a combinatorial proof for the following: Prove the identity $$\sum_{i=0}^{k}{m+k-i-1 \choose k-i}{n+i-1 \choose i}={m+n+k-1 \choose k}$$ Hint: use idea of "selection with repetition". ...
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### Proof that binomial coefficient is a natural number [duplicate]

Possible Duplicate: Proof that a Combination is an integer What is the proof that the binomial coefficient is a natural number? $$k\ge0,n\ge k \implies {n \choose k} \in N,$$ I guess ...
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### Maximum term of (a + b) ^ n

I would like a demonstration of the fact below. Being given real numbers a and b (nonzero) and a positive integer n, the order p, that occupies the maximum term (in absolute value) of the ...
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### Another sum involving binomial coefficients.

Let both $a$,$b$ and $\theta$ be real numbers not equal to a negative integer. Let $m$ be a positive integer. I have shown that the following equality holds: \begin{eqnarray} ...
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### Prove Maximum term in the expansion.

How should I prove the maximum term in the expansion of $(x+a)^n$ where $ax>0$ is the term $C(n,r)x^{(n-r)}a^r$ for which $r= \left[\cfrac{(n+1)}{(n/a)+1} \right]$ ?
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### Proof of a binomial identity $\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$

Prove that $$\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$$ The exercise provides the following hint: $\,\,\displaystyle{n \choose k}={n\choose n-k}$. Any help?
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### Solutions to $\binom{n}{5} = 2 \binom{m}{5}$

In Finite Mathematics by Lial et al. (10th ed.), problem 8.3.34 says: On National Public Radio, the Weekend Edition program posed the following probability problem: Given a certain number of ...
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### Strehl identity for the sum of cubes of binomial coefficients

In 1993 Strehl showed that $$\sum_k\binom nk^3=\sum_k\binom nk^2\binom{2k}n.$$ I’m interested in a combinatorial proof. Upd (Jan '14). Maybe the original question was too restrictive — I'm now ...
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### Evaluate a finite sum with four factorials

Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate $$\sum^{n}_{i\mathop{=}0}\frac{1}{n+k+i}\cdot\frac{(m+n+i)!}{i!(n-i)!(m+i)!}$$ Any hints? I'm stuck on ...
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### How to evaluate $\sum\limits_{k=0}^{n} \sqrt{\binom{n}{k}}$

Can we find $$\sum_{k=0}^{n} \sqrt{\binom{n}{k}} \quad$$ This problem asked me my friend about a year ago, but I didn't know how to attack problem. Now, I am interesting in solution. Any suggestion? ...
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### proof of formula and calculation sum

Show that following formula is true: $$\sum_{i=0}^{[n/2]}(-1)^i (n-2i)^n{n \choose i}=2^{n-1}n!$$ Using formula calculate $$\sum_{i=0}^n(2i-n)^p{p \choose i}$$
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### Is there a direct proof of this lcm identity?

The identity $\displaystyle (n+1) \text{lcm} \left( {n \choose 0}, {n \choose 1}, ... {n \choose n} \right) = \text{lcm}(1, 2, ... n+1)$ is probably not well-known. The only way I know how to prove ...
### Binomial Sum Related to Fibonacci: $\sum\binom{n-i}j\binom{n-j}i=F_{2n+1}$
How would I prove $$\sum\limits_{\vphantom{\large A}i\,,\,j\ \geq\ 0}{n-i \choose j} {n-j \choose i} =F_{2n+1}$$ where $n$ is a nonnegative integer and $\{F_n\}_{n\ge 0}$ is a sequence of ...