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

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2
votes
1answer
49 views

Simplify the expression $\binom{n}{0}+\binom{n+1}{1}+\binom{n+2}{2}+\cdots +\binom{n+k}{k}$ [duplicate]

Simplify the expression $\binom{n}{0}+\binom{n+1}{1}+\binom{n+2}{2}+\cdots +\binom{n+k}{k}$ My attempt: Using the formula $\binom{n+1}{k}=\binom{n}{k}+\binom{n}{k-1}$ $\binom{n}{0}+\binom{n+1}{1}+\...
-1
votes
2answers
43 views

want to check answer given in my book is correct or not [duplicate]

The problem is to find coefficient of $x^n$ using binomial theorem for rational index in the expansion of $$\frac{1}{1-x+x^2-x^3}.$$ In my book the answer is given as $$\frac14+\frac{n+1}{2}+\frac{(-...
-1
votes
1answer
63 views

Tricky question involving binomial expansion [on hold]

For a fixed m, what is the highest power of $2$ that divides $[(\sqrt3 +1)^m]+1$? where $[x]$ denotes the greatest integer less than or equal to $x$. I have no clue how to proceed.
0
votes
1answer
20 views

How do I find the terms of an expansion using combinatorial reasoning?

From my textbook: The expansion of $(x + y)^3$ can be found using combinatorial reasoning instead of multiplying the three terms out. When $(x + y)^3 = (x + y)(x + y)(x + y)$ is expanded, all ...
3
votes
2answers
101 views

Find the value of $\sum_{r=0}^{\left\lfloor\frac{n-1}3\right\rfloor}\binom{n}{3r+1}$

Show that $$\binom{n}{1}+\binom{n}{4}+\binom{n}{7}+\ldots=\dfrac{1}{3}\left[ 2^{n-2} + 2\cos{\dfrac{(n-2)\pi}{3}}\right]$$ My solution:- $$(1+x)^n=\binom{n}{0}+\binom{n}{1}x+\binom{n}{2}x^2+\...
1
vote
3answers
34 views

Problem related to series of binomial coefficients

Problem related to series of binomial coefficients in which each term is a product of two binomial coefficients. In this question: Prove that $$\binom{n}0^2+\binom{n}1^2+\ldots+\binom{n}n^2=\...
0
votes
1answer
14 views

Prove that: ${^{n}\mathrm{C}_{k}} = {^{n-1}\mathrm{C}_{k-1}}+{^{n-1}\mathrm{C}_{k}}$ [duplicate]

Question asks to prove: ${^{n}\mathrm{C}_{k}} = {^{n-1}\mathrm{C}_{k-1}}+{^{n-1}\mathrm{C}_{k}}$ My Steps: $$\begin{align*}\frac{(n-1)!}{(n-k-2)!(k-1)!} + \frac{(n-1)!}{(n-k-1)!(k)!} & = \...
0
votes
1answer
58 views

Find the coefficient of $x^n$ simply by using binomial theorem [on hold]

Find the coefficient of $x^n $ in the expansion of $\frac{1}{1-x+x^2-x^3}$.
0
votes
2answers
24 views

Formation of Teams in Permutation and Combination

A class has $n$ students , we have to form a team of the students including at least two and also excluding at least two students. The number of ways of forming the team is My Approach : To include ...
1
vote
1answer
24 views

Need help with inductive proof of Binomial Theorem

I'm new to math and trying to learn about the Binomial Theorem, by following this tutorial. I got stuck trying to read the Induction Proof. They give an example of using the Sum notation: $$ (x + y)^...
1
vote
0answers
35 views

Probability question, can I reset the window or not

There is a wall street banker. The banker invests in a kind of share called as options. The main features of this share is as follows: You make a bet with a specified amount of information as to ...
1
vote
3answers
43 views

Identical Obects in Permutation and combination

There are $2$ identical white balls , $3$ identical red balls and $4$ green balls of different shades. The number of ways in which the balls can be arranged in a row so that at least one ball is ...
-4
votes
1answer
38 views

Binomial Expansion: precalculus

In the expansion $(1 +px)(1 +qx)^4$ in ascending powers of $x$ , the coefficient of the $x$ term is $-5$ and there is no $x^2$ term. Find the value of $p$ and $q$. My attempted answer 1) I ...
0
votes
2answers
76 views

Is there a formula for the binomial expansion of $(a-b)^n$?

Like there is a formula for the binomial expansion of $(a+b)^n$ that can be neatly and compactly be written as a summation, does there exist an equivalent formula for $(a-b)^n$ ?
2
votes
1answer
46 views

If $f(n) =\displaystyle\sum_{r=1}^{n}\Biggl(r^n\Bigg(\binom{n}{r}-\binom{n}{r-1}\Bigg) + (2r+1)\binom{n}{r}\Biggr)$, then what is $f(30)$?

Please give me hints on how to solve it. I tried 2-3 methods but it doesn't go beyond two steps. I am out of ideas now. Thank you
1
vote
0answers
89 views

Ellipsoidal harmonics - A Series expansion for Lame functions of the second kind

Intro to skip In the theory of ellipsoidal harmonics, Lame functions of the second kind $F_n$ arise as the second linearly independent solution (the first being Lame functions of the first kind $E_n$)...
0
votes
4answers
104 views

$ z = 1 + 2i $ - Prove that $ z^n \notin \mathbb{R} $ [duplicate]

$$ z = 1 + 2i \ (complex \ number) \\ z^n = a_n + b_ni \ (a_n, b_n \in \mathbb{Z}, n \in \mathbb{N}^*) \\ We \ know \ that: \ b_{n+2} - 2b_{n+1} + 5b_n = 0 \\ a_{n+1}=a_n-2b_n \\ b_{n+1}=b_n+2a_n $$ ...
1
vote
3answers
50 views

$ z^n = a_n + b_ni $ Show that $ b_{n+2} - 2b_{n+1} + 5b_n = 0 $ (complex numbers)

$$ z = 1+2i \ (complex \ number) \\ z^n = a_n + b_ni \ \ \ (a_n, b_n \in \mathbb{Z}, n \in \mathbb{N}^*) $$ Prove that $ b_{n+2} - 2b_{n+1} + 5b_n = 0$ How can I solve this? Thank you! EDIT: Or ...
-1
votes
0answers
21 views

General Binomial Theorem for Noncommutative Elements

I want to know how to expand a binomial $ (A + B)^\alpha $ where $\alpha \in \mathbb{R} $ and $ [A,B] = C \neq 0 $. A very similar question was asked on Math Overflow (http://mathoverflow.net/...
6
votes
3answers
121 views

Compute $\sum\limits_{k=0}^{100}\frac{1}{(100-k)!(100+k)!}$

$$\sum_{k=0}^{100}\frac{1}{(100-k)!(100+k)!}$$ My work $$\sum_{k=0}^{100}\frac{2n!}{(2n!)(n-k)!(n+k)!}$$ $$\sum_{k=0}^{100}\frac{^{2n}C_{n-k}}{(2n!)}$$ $$\sum_{k=0}^{100}\frac{^{2n}C_{n+k}}{(2n!)}$$...
0
votes
3answers
54 views

Prove that $a_n-10a_{n-1}+a_{n-2}=0$.

Let $a_n = (5+2\sqrt{6})^n+(5-2\sqrt{6})^n$. Prove that $a_n-10a_{n-1}+a_{n-2}=0$. I think this depends on whether $n$ is even or odd so in the case $n$ is even we have $a_n = 2(\binom{n}{0}5^n+\...
2
votes
2answers
60 views

Finding maximum value of ${n \choose r}$ for given value of n [duplicate]

While I was solving some binomial theorem chapter questions I encountered many questions which asked me me to find maximum value of ${n \choose r}$ for given value of n. Example: Find n for which $...
6
votes
4answers
123 views

Express $1 + \frac {1}{2} \binom{n}{1} + \frac {1}{3} \binom{n}{2} + \dotsb + \frac{1}{n + 1}\binom{n}{n}$ in a simplifed form

I need to express $$1 + \frac {1}{2} \binom{n}{1} + \frac {1}{3} \binom{n}{2} + \dotsb + \frac{1}{n + 1}\binom{n}{n}$$ in a simplified form. So I used the identity $$(1+x)^n=1 + \binom{n}{1}x + \...
0
votes
4answers
48 views

Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n - 1}$ using the binomial theorem

I'm trying to prove that \begin{equation} \sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n - 1} \end{equation} with the Binomial Theorem. I know that the B.T. states that \begin{equation} (x + y)^n = ...
-1
votes
1answer
37 views

Reducing Binomial Summation [closed]

How can I reduce this summation into this $$\frac{1}{2}(1+\left(1/3\right)^{50})$$ The problem comes from the 1992 AHSME Test (problem 29)
1
vote
0answers
39 views

A generalization for Binomial Theorem, Leibniz General Rule other like functions.

$n$ & $m :=$ any value in $\{0,1,2\ldots\}$ $\Omega$ & $\beta :=$ any value in $\{1,2,3\ldots\}$ [1] If there is a function $F_\beta$ such that for some value $\Omega$ and some function $T_{...
2
votes
1answer
46 views

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. \begin{equation} \frac{1}{\left[\sum_{k=0}^\infty a_kt^{n-2k}\right]^2} \end{equation} as a series ...
2
votes
2answers
47 views

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 $(x^5-1)^{199}(x-1)^{2}$. ...
1
vote
2answers
40 views

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 ...
0
votes
1answer
6 views

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 $(x+y)^2$ ...
1
vote
2answers
46 views

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 ...
1
vote
1answer
27 views

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 ...
0
votes
3answers
111 views

Finding the coefficient of $x^7$ in the expansion $(1 + x)^{23}$

By definition, the Binomial Theorem states: $$(x+y)^n = {n\choose 0}x^n + {n\choose 1}x^{n-1}y + {n\choose 2}x^{n-2}y^2 + \cdots + {n \choose {n-1}}xy^{n-1} + {n \choose n}y^n$$ For any $x,y\in\...
3
votes
1answer
55 views

On the proof of Lucas' theorem

Lucas theorem states that Let $m,n$ be two natural numbers, $p$ be a prime. Suppose that $m, n$ admit the following base $p$ representation $$m=m_0+m_1p+\cdots+m_sp^s,\qquad n=n_0+n_1p+\cdots+n_sp^...
3
votes
1answer
30 views

Complex Numbers and Binomial Expansion

I was able to show the above by equating the solution of $cos \ 5\theta=0$ (which gives $\pi/10$) and the solution of ${16cos}^5\theta - {20cos}^3\theta+5cos\theta = 0$ (which is what you get when you ...
12
votes
3answers
181 views

Intuitive ways to get formula of binomial-like sum

Is there an intuitive way, though I am not sure how to find a conceptual proof either, to establish the following identity: $$\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$$ for all natural ...
1
vote
5answers
100 views

The coefficient of $x^{100}$ in the expansion of $(1-x)^{-3}$.

Please solve it and tell me the technique so that I can solve it in examination in multiple choice questions.
1
vote
2answers
43 views

Total no of terms in the expansion of $(x+z)^{50} +(x-z)^{50}$

Please help me in solving the question and also tell me the technique so that I can easily solve it when similar question come in the form of Multiple choice questions MCQS.
0
votes
0answers
15 views

Prove that the binomial theorem for complex $r$ reduces to the integer $r$ formula

According to wikipedia the binomial theorem states that if $|x|>|y|$ and $r$ complex $$ (x+y)^r=\sum_{k=0}^{\infty}\frac{(r)_k}{k!}x^{r-k}y^k $$ where is the $$ (r)_k=\frac{\Gamma(r+1)}{\Gamma(r-...
1
vote
1answer
48 views

Inclusion-Exclusion principle and finite product identity

On this page, there is a proof that uses the inclusion-exclusion principle to provide a formula for the value of Euler's totient function. I would be grateful if someone could explain the reasoning ...
0
votes
1answer
46 views

Sum reminiscent of $(1+x)^N$ (binomial theorem)

I stumbled upon this sum while working on my thesis: $$\sum_{k=0}^N \binom{N}{2k} x^k$$ I know that $$\sum_{k=0}^N \binom{N}{k} x^k = (1+x)^N$$ But when it comes to the sum above I'm lost. Is ...
1
vote
1answer
39 views

Can we apply binomial theorem for $\quad(a+b)^\ell\quad$ if $\ell\;$ irrational.

Let be$\quad a,b\;\in\mathbb R\quad, \ell\;\in\mathbb {(R\backslash Q)} \quad $ ($\ell:$irrational) Can we apply binomial theorem for $\quad(a+b)^\ell$
2
votes
1answer
24 views

How to expand an expression containing $x^2$ by binomial theorem

I understand that the coefficient of, say $x^8$, in the expansion of $(1+x)^{10}$, would be ${10 \choose 8}$, but what about an expression like $(1+x^2)^{10}$? Would I have to square root the ${10 \...
-1
votes
1answer
26 views

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 ...
3
votes
1answer
83 views

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 ...
0
votes
3answers
76 views

Proof that combinations are equal to coefficients in the binomial expansion

Let $n\in N$, $k\in Z$, $o\leq k \leq n$. Define $C^{n}_k$ as the coefficient of $x^{n-k}y^k$ in the expansion of $(x+y)^n$ $$(x+y)^n= \sum^{n}_{k=0} C^{n}_k x^{n-k}y^k$$ Prove that ${C^{n}_{k}}={...
0
votes
1answer
37 views

Binomial expansion of $(1 - w z^{-1})^{-1}$ being $\sum_{l=0}^{\infty} w^l z^{-l}$

I am reading "Neural Networks and Learning Machines", Third Edition, Simon Haykin and at the beginning I came across at something that perplexes me. I'll quote a whole page because some contexts may ...
1
vote
2answers
84 views

Find $\sum\limits_{r=0}^n(-1)^r\binom{n}{r}^{-1}$ for $n$ even [duplicate]

If $n$ is an even natural number, then find $$\sum_{r=0}^n \left(\frac{(-1)^r}{\binom{n}{r}}\right)$$ I tried to solve the question using conventional method, by trying to use calculus, but I ...
2
votes
1answer
47 views

Why does the binomial theorem for negative numbers never terminate

Why does $(a+bx)^n$ never terminate if you use negative numbers or fractions for n? Surely $(a+bx)^{-2}=\frac{1}{a^2+2abx+b^2x^2}$ and not an infinite series? Thanks
1
vote
1answer
31 views

Finding coefficient of $x^n$ in this series

I was doing an assignment question, I came across this: I understand everything (I know the theorem used in the answer), but I don't get how solution switched from first "i.e." to second "i.e.". I ...