# Tagged Questions

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### Squared binomial paradox?

When you square this $$(5-2)^2$$ you will get 49 $$5^2 - 2 * 5 * (-2) + (-2)^2$$ $$25 + 20 + 4 = 49$$ but if you do it like this (5-2) * (5-2) you will get 9 $$5(5-2) - 2(5-2)$$ $$25-10-10+4$$ ...
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### simplify $(a_1 + a_2 +a_3+… +a_n)^m$

How to simplify this best $(a_1 + a_2 +a_3+... +a_n)^m$ for $m=n, m<n, m>n$ I could only get $\sum_{i=0}^{m}\binom{m}{i}a_i^i\sum_{j=0}^{m-i}\binom{m-i}{j}a_j ...$
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### Why is the binomial coefficient related to the binomial theorem?

The binomial coefficient basically provides the number of ways to choose a set of $k$ from $n$ sets. To me, it can be considered the number of unique ways to pick $k$ amount of "cards" from a deck of ...
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### Showing that $\lceil (\sqrt{3} + 1)^{2n} \rceil$ is divisible by $2^{n+1}$.

I have a question which has fluxommed me and my pals for the past few days. Any help or solution is welcome Show using Binomial theorem that the integer just after $(3^{1/2} + 1)^{2n}$ is divisble ...
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### The sum of the coefficients of $x^3$ in $(1-\frac{x}{2}+\frac{1}{\sqrt x})^8$

I know how to solve such questions when it's like $(x+y)^n$ but I'm not sure about this one: In $(1-\frac{x}{2}+\frac{1}{\sqrt x})^8$, What's the sum of the coefficients of $x^3$?
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### Hard elementary combinatorics problem

How does one compute (without brute force) the smallest integer $n$ such that $\binom{2n}{1}(-3)^0 + \binom{2n}{3}(-3)^1 + \binom{2n}{5}(-3)^2 + \cdots + \binom{2n}{2n-1}(-3)^{(n-1)} = 0$?
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### Binomial theorem application

I have a question about the bonomial theorem, and in specifically, a question that I want help on. I have worked out the answer, but by manually expanding each and every alternative. However, I ...
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### Calculate the expansion of $(x+y+z)^n$

The question that I have to solve is an answer on the question "How many terms are in the expansion?". Depending on how you define "term" you can become two different formulas to calculate the terms ...
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### Water and wine mixing problem

This is a well-known problem involving a water barrel and a wine barrel, described here. The trick to solving the puzzle is that one need not make the calculations for each stage of the liquid ...
### How to prove that $\sum\limits_{i=0}^p (-1)^{p-i} {p \choose i} i^j$ is $0$ for $j < p$ and $p!$ for $j = p$
Let $p \in \mathbf{N}$. I don't know how to prove that $$\sum_{i=0}^p (-1)^{p-i} {p \choose i} i^j=0 \textrm{ for } j \in \{0,\ldots,p-1\},$$ and $$\sum_{i=0}^p (-1)^{p-i} {p \choose i} i^p=p!$$ ...
### How can I compute $\sum\limits_{k = 1}^n \frac{1} {k + 1}\binom{n}{k}$?
This sum is difficult. How can I compute it, without using calculus? $$\sum_{k = 1}^n \frac1{k + 1}\binom{n}{k}$$ If someone can explain some technique to do it, I'd appreciate it. Or advice using ...