# Tagged Questions

For questions related to the binomial theorem, which describes the algebraic expansion of powers of binomials.

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### Is it possible to express the inverse of a polynomial as a series?

Is it possible to express the multiplicative inverse of a polynomial in descending powers of n i.e. $$\frac{1}{\left[\sum_{k=0}^\infty a_kt^{n-2k}\right]^2}$$ as a series ...
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### Coefficient of $x^{103}$ in the following multinomial expansion.

What is the coefficient of $x^{103}$ in the expansion of $$(1+x+x^2+x^3+x^4)^{199}(x-1)^{201}$$ ?. The answer is an integer between $0-9$. So I wrote the given expression as $(x^5-1)^{199}(x-1)^{2}$. ...
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### Binomial theorem question. Find the value of the constant $k$

$$\left[(k+x)\left(2-\frac{x}{2}\right)\right]^6$$ where the coefficient of $x^{2}$ is $84$.Find the value of the constant $k$. I tried to expand the equation but got a equation of degree 6 for some ...
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### Show that $x(1- (\frac{R_1-R_2}{x})^{2})^{0.5}$ = x – $\frac{(r1 – r2)^2}{2x}$

Show that $x(1- (\frac{R_1-R_2}{x})^{2})^{0.5}$ = x [1 – {0.5 $\frac{(r1 – r2)}{x}$ + … ] = x – $\frac{(r1 – r2)^2}{2x}$ by using binomial theorem it was mentioned that binomial theorem is $(x+y)^2$ ...
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### Proving $(w-1)^m$ is purely imaginary.

I'm having trouble trying to prove this: Let $m\in \mathbb Z$, m even and $w\in\mathbb C$ a primitive $2m$-th root of unity. Prove that $(w-1)^m$ is purely imaginary. What I've tried to do so ...
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### Probability of Getting a Yahtzee of Fives Given Two Fives

(The following problem is from MAML, Meet 3, Round 1, December 2012, Problem 3.) In the game of Yahtzee one has a chance to get Yahtzee (5 of the same kind, such as 5 sixes) in the throw of 5 ...
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### Problem to prove for all even integers

Prove that $5^n = 3^n + \dfrac{16n (3^{n-2})}{2} + \dfrac{256n (n-2) 3^{n-4}}{8}+ ....+ 4^n$ for all even integers. I tried finding a pattern, but was unable to do so. Any help would be ...
### Find $\sum_{k=1}^n \binom{2n-k}{n}(-1)^k$
Is there closed form for $\sum_{k=1}^n \binom{2n-k}{n}(-1)^k$? I got above expression for a counting exercise. I wonder that it might have the closed form but I am not sure yet. Can anyone have any ...