Questions tagged [binomial-theorem]

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

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What does Euler mean when he says "Let the arc z be infinitely small ; there will be sin.z=z and cos.z=1..."?

I am referring to, specifically the sin.z=z and cos.z=1. I am reading Euler's Introductio In Analysin Infinitorium, Vol. I, Ch. VIII and he jumps from binomials to ...
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How to use Laplace's Method to obtain asymptotic approximation

I'm trying to use Laplace's method to expand this integral: \begin{eqnarray*} I = &\int_{0}^{+\infty}\left(1+\frac{x}{n}\right)^n e^{-x}\,dx\end{eqnarray*} as $n \to \infty$. I know $I \sim {\sqrt{...
2 votes
7 answers
88 views

Prove that $ \sum_{i=0}^n {n \choose i} \cdot (-1)^i \cdot i^n =(-1)^n n!$ and $ \sum_{i=0}^n {n \choose i} \cdot (-1)^i \cdot i^k = 0$ for $k<n$

For natural any number $n$, prove that $\sum_{i=0}^n{n \choose i}\cdot (-1)^i \cdot i^n =(-1)^n n!$ and for whole number $k<n,\,\sum_{i=0}^n {n \choose i} \cdot (-1)^i \cdot i^k =0$. I know that $\...
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2 votes
2 answers
56 views

Prove that $\Sigma_{i=0}^{n-1} (\Sigma_{j=i+1}^n(j(^nC_i) + i(^nC_j))) = n^22^{n-1}$ [duplicate]

I have tried setting $$P=\Sigma_{i=0}^{n-1} (\Sigma_{j=i+1}^n(j(^nC_i) + i(^nC_j))$$ then $$P=\Sigma_{i=0}^{n-1} (\Sigma_{j=i+1}^n((n-j)(^nC_i) + (n-i)(^nC_j))$$ now adding them both $$2P=\Sigma_{i=0}^...
0 votes
0 answers
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On the sum: $\sum_{i = 0}^n i {n \choose i} = n 2^{n-1}$ [duplicate]

I need to prove that: $$ \sum_{i = 0}^n ~ i ~ {n \choose i} = n ~ 2^{n-1} $$ Been stuck for a while now, will appreciate any help. Thanks in advance.
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The sum of the squares of coefficients on a row of Pascal triangle is equal to $\binom{2n}{n}$ [duplicate]

I have an identity: $$ \binom{2n}{n}=\binom{n}{0}^2+\binom{n}{1}^2+\binom{n}{2}^2+...+\binom{n}{n}^2 $$ My first thought to solve this identity is to use the binomial theorem. The binomial theorem ...
0 votes
1 answer
52 views

How to show that $12^{12} + 9^{9}$ can be divided by $15$ using Binomial Theorem? [duplicate]

How to show that $12^{12} + 9^{9}$ can be divided by $15$ using Binomial Theorem? I don't know where to start, given the specific method needed. Since both powers are divisible by 3, I know that the ...
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1 vote
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Find all rational terms of $(\sqrt{2}+\sqrt[3]{3})^{100}$ [duplicate]

The title is the problem: how many rational terms are there in $(\sqrt{2}+\sqrt[3]{3})^{100}$? I am very close to the answer and intuitively I know the answer, but I've been struggling for an hour ...
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I would like to prove that $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = \sum_{k=0}^{+\infty}\frac{1}{k!}.$ [duplicate]

I would like to prove that $$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = \sum_{k=0}^{+\infty}\frac{1}{k!}.$$ Among the several attempts I've performed to prove the previous, I report the ...
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1 vote
1 answer
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Sum with binomial coefficient using identity

I want to prove: $\displaystyle \sum_{k=0}^n (-1)^k \binom{x}{k} = (-1)^n \binom{x-1}{n}$ using: $(1-z)^x \cdot \frac{1}{1-z} = (1-z)^{x-1}$ I know how to do it with induction but i somehow can't ...
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3 votes
2 answers
103 views

Find the value of three expressions

Consider $$(1+x+x^2)^n=\sum_{r=0}^{2n}a_rx^r$$ where $a_0,a_1,a_2,\cdots,a_{2n}$ are real numbers and $n$ is a positive integer. Then find the value of $$\sum_{r=0}^{n-1}a_{2r}$$ And $$\sum_{r=1}^...
3 votes
2 answers
67 views

Solving $f\left( n \right) = \sum\limits_{i > j \ge 0} {{}^{n + 1}{C_i}{}^n{C_j}} $

My approah is as follow Let $f\left( n \right) = \sum\limits_{i > j \ge 0} {{}^{n + 1}{C_i}{}^n{C_j}} $ $f\left( n \right) = \sum\limits_{i = i}^n {{}^{n + 1}{C_i}} \sum\limits_{j = 0}^i {{}^i{C_j}}...
6 votes
4 answers
184 views

Prove a binomial identity: $\sum_{i=1}^n i \binom{2n}{n-i}=\frac12(n+1) \binom{2n}{n-1}$

I want to prove the product of even/odd power with a combinatorial number: \begin{aligned} \sum_{i=1}^n i \binom{2n}{n-i}&=\frac12(n+1) \binom{2n}{n-1}, \\ \sum_{i=1}^n i^2 \binom{2n}{n-i}&...
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1 vote
1 answer
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Trouble with understanding the proof of the contribution of the principle part on the coefficients of the function

$PP(f;z_0)$ represents the principle part of $f(z)$ at pole $z_0$ $r$ represents the order of the pole My confusion arises in the third row where the paper gets that $$\left(1-\frac{z}{z_0}\right)^{-...
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3 votes
0 answers
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If $A=\lim_{n\to\infty}\bigg(\prod_{k=0}^{n} {n\choose k}\bigg) ^{\frac{1}{n(n+1)}}$. Then Find $A$ [duplicate]

If $$A=\lim_{n\to\infty}\bigg(\prod_{k=0}^{n} {n\choose k}\bigg) ^{\frac{1}{n(n+1)}}$$. Then Find $A$ My Approach: $\prod_{k=0}^{n} {n\choose k}=\prod_{k=0}^{n}\dfrac{n!}{k!(n-k)!}=\dfrac{(n!)^{n+1}}{...
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1 vote
3 answers
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Assuming x is small, expand $\frac{\sqrt{1-x}}{\sqrt{1+2x}}$ up to and including the term in $x^{2}$

I have tried this many times but can't quite land on the correct answer. The correct answer: $1-\frac{3x}{2}+\frac{15x^{2}}{8}$ These are the steps I took: Re wrote it as: $\left ( 1-x \right )^{\...
2 votes
1 answer
52 views

Show that all the zeros of $(az+1)^m-b (z+1)^m$ lie in $|z|<1,$ where $a>1$ and $|b|\leq 1.$

If $f(z)=(z+1)^m$ then $f(az)=(az+1)^m,$ where $m\in\mathbb{N}.$ I can prove that $|f(z)|<|f(az)|$ for $|z|=1$ and $a>1.$ Therefore, by Rouche's theorem, all the zeros of $(az+1)^m-b (z+1)^m$ ...
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1 vote
1 answer
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How can I estimate the number of repeated selections

I have a very challenging problem that I cannot find a way to solve without python simulation. Given a dataset of size X (very large number), we want to select H entries from X without replacement and ...
0 votes
1 answer
104 views

Sum of coefficients in the expansion of $(2x+3y-2z)^n$.

Sum of coefficients in the expansion of $(2x+3y-2z)^n$ is $2187$ then the greatest coefficient in the expansion of $(1+x)^n$. I put $x=y=z=1$ in the expansion. So, sum of coefficients became equal to ...
1 vote
0 answers
52 views

Understanding the generalised binomial theorem

In lecture we wrote the following: Generalised Binomial Theorem: For any $\alpha \in \mathbb{C}$ holds $$(1+x)^\alpha = \sum_{k=0}^\infty \binom{\alpha}{k} x^k$$ where $x$ is a formal variable or $x \...
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Factoring out a coefficient from binomial

I watched a youtube video (I cannot find it anymore) however, the author showed that the following binomial equation could be factored (If I remembered it correctly.) $$\binom{n}{n+2k}=\binom{n}{n+k}\...
2 votes
3 answers
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How to prove the binomial identity $\binom{n + 1}{a + b + 1} = \sum_{k = 0}^n \binom{k}{a}\binom{n - k}{b}$

Prove the identity: $$\binom{n + 1}{a + b + 1} = \sum_{k = 0}^n \binom{k}{a}\binom{n - k}{b}$$ So far I understand the left side represents how many ways there are picking a+b+1 elements from a set (...
6 votes
0 answers
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Verify proof: $a^{n}-b^{n} = (a-b) \sum\limits_{k=0}^{n-1} a^{k}b^{n-1-k}$

A short disclaimer: I do know this question has been asked multiple times here and several answers (including combinatorics) have been given already. However, among all these posts, I did not find ...
0 votes
1 answer
44 views

Hi I am confused how to solve this question [closed]

Hi I have been trying this question for days but cannot get to the answer. This question seems to be a lot tough. I tried many ways but cannot reach the answer. It will be a humble request if anyone ...
0 votes
1 answer
186 views

How many collected terms are in the expansion of $(x+y+z)^{10} (w+x+y+z)^2$?

How many collected terms are in the expansion of $(x+y+z)^{10} (w+x+y+z)^2$? Hi, I'm trying to solve this problem as study material for discrete mathematics and I'm not quite sure how. I got 235 terms ...
3 votes
2 answers
99 views

About a sum with a combinatorial coefficient

I'm reading the Wikipedia article on analytic continuation and I got stuck in the last step of the worked example. How to get the following result? $$\sum_{n=0}^{\infty}(-1)^n{n\choose k}(a-1)^{n-k}=(-...
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4 votes
2 answers
204 views

$\binom{54}{5}+\binom{49}{5}+\binom{44}{5}+\cdots+\binom{9}{5}$

How to calculate the sum $$\binom{54}{5}+\binom{49}{5}+\binom{44}{5}+\cdots+\binom{9}{5}$$ I wrote this as $$\sum_{r=2}^{11}\binom {5r-1}{5}$$ $$=\frac{1}{120}\sum_{r=2}^{11}(5r-1)(5r-2)(5r-3)(5r-4)(...
5 votes
3 answers
275 views

Which number is larger? Using Binomial Theorem

Which expression is larger, $$ 99^{50}+100^{50}\quad\textrm{ or }\quad 101^{50}? $$ Idea is to use the Binomial Theorem: The right hand side then becomes $$ 101^{50}=(100+1)^{50}=\sum_{k=0}^{50}\binom{...
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1 vote
1 answer
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Probability of flipping four coins

Flip 4 coins, when you see a head, you get \$1. What is the expected return of this game? What if you can reflip (meaning reflip all coins) as many times as you want? What if you have to pay $1 each ...
4 votes
1 answer
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Binomial theorem-respecting polynomials

I'll call a family of polynomials "binomial theorem-respecting" if they satisfy: $$ f_m(x+k) = \sum_{i=0}^m {m \choose i} k^{m-i} f_i(x) $$ For example, the family of polynomials $f_m(x) = x^...
-1 votes
1 answer
49 views

Problem with binomial sum [closed]

I need to calculate the sum:$$\sum_{k=0}^n \frac{(2k+1)}{k+1}\binom{n}{k}$$ but can't find a right technique to do so. Could anyone give me a hit?
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1 vote
1 answer
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Double summation over binomial coefficients

I tried to solve $\displaystyle \sum_{j=0}^{n} \sum_{i=j}^{n} \binom{n}{i} \binom{i}{j}$. Unfortunately, I cannot succeed in doing so. It comes with a hint that says: "Interchange the sums and ...
-1 votes
2 answers
37 views

Prove that the next integer to $(\sqrt3+1)^{2n}$ is $(\sqrt3+1)^{2n}+(\sqrt3-1)^{2n}$

How the integer next to $(\displaystyle\sqrt3+1)^{2n}$ is $$(\sqrt3+1)^{2n}+(\sqrt3-1)^{2n}$$ In my opinion, the next integer should be $$(\sqrt3+1)^{2n}-(\sqrt3-1)^{2n}+1$$ I saw an answer on math....
1 vote
0 answers
42 views

Binomial collisions algorithm

I am looking for an efficient algorithm to find collisions in the pascal triangle with repeats $\ge 4$. Definition of a collision: let (n, k, m, l) with 2<=k<= (1/2)*n and 2<=l<=(1/2)*m. A ...
0 votes
0 answers
16 views

Unable to prove binomial based identity related to succesive difference between nth power of numbers? [duplicate]

https://youtube.com/shorts/I0WF5-a2B7g?feature=share I was trying to prove a mathematical identity that I come up with after watching the above video. $$\sum_{k=0}^n (-1)^k \binom{n}{k}(n-k+M)^n =n! \\...
3 votes
0 answers
56 views

Combinatorics proof of Abel's binomial theorem

The Abel's identity which is given with Abel polynomials: $$\sum_{k=0}^n\binom{n}{k}a(a+kz)^{k-1}b(b+(n-k)z)^{n-k-1}=(a+b)(a+b+nz)^{n-1}$$ A possible proof is to simply calculate coefficient of $x^n$ ...
0 votes
1 answer
48 views

Binomial formula for matrix $(\mathbf{I}+\mathbf{A})^{-1/2}$.

I am wondering if the following binomial expansion for the matrix is correct: $$ (\mathbf{I}+\mathbf{A})^{-1/2}=\sum_{n=0}^{\infty}\left(\begin{array}{c} -1/2 \\ n \end{array}\right) \mathbf{A}^n, $$ ...
0 votes
1 answer
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Show with the help of binomial theorem that these two equations are equal?

Show with the help of binomial theorem that these two expression are equal for $n\ge 0$ then this $$ \sum_{k=0}^n \binom n k x^k (2+x)^k = \sum_{k=0}^{2n} \binom {2n} k x^k $$ I don’t know how to do ...
0 votes
0 answers
54 views

Equal coefficients in binomial expansions

Find all positive integers $n$ so that there are at least two coefficients in the expansion of $(7x+1)^n$ are equal. I worked out the special case that two "consecutive coefficients" are ...
-2 votes
1 answer
107 views

Find the last $2$ digits of $3^{2024}$ using binomial expansion [duplicate]

Find the last two digits of $$3^{2024}$$ I can easily do this question (see below) but is there a way of doing it by binomial expansion? I tried to expand the expression to a few terms but in vain. ...
0 votes
1 answer
124 views

$\binom{n+1}{2} + 2 \left ( \binom{2}{2} + \binom{3}{2} + \dots + \binom{n}{2} \right ) = 1^2+2^2+\dots+n^2.$

Is there a combinatorial proof of the following identity: $$\binom{n+1}{2} + 2 \left ( \binom{2}{2} + \binom{3}{2} + \dots + \binom{n}{2} \right ) = 1^2+2^2+\dots+n^2.$$ Note: using algebraic ...
1 vote
2 answers
90 views

Prove $\binom{n}{0}$ + $\binom{n}{2}$ + $\binom{n}{4} + ...+\binom{n}{n-1}=\binom{n}{1}+\binom{n}{3}+\binom{n}{5}+...+\binom{n}{n}$

I want to prove that $\binom{n}{0}$ + $\binom{n}{2}$ + $\binom{n}{4} + ...+\binom{n}{n-1}=\binom{n}{1}+\binom{n}{3}+\binom{n}{5}+...+\binom{n}{n}$ for all odd numbers $n \ge 1$ I see that $\binom{n}{0}...
1 vote
0 answers
49 views

Let $p$ be an odd prime and let $n$ be a positive integer. Use the Binomial Theorem to show that $(1+p)^{p^{n-2}}\not\equiv1$ (mod $p^n$).

The Problem: Let $p$ be an odd prime and let $n$ be a positive integer. Use the Binomial Theorem to show that $(1+p)^{p^{n-2}}\not\equiv1$ (mod $p^n$). Source: Abstract Algebra, $3^{rd}$ edition, ...
2 votes
1 answer
61 views

Having difficulties with pendulum theory and percentage error homework problem.

The pendulum theory: $$t=2 \pi \sqrt{l/g},$$ where $t$ is the time of period, $L$ is the length of the pendulum, $G$ is the acceleration due to the gravity (~9.81 m/s²). Calculate the expected ...
-2 votes
2 answers
122 views

how to calculate the value $0.9^{31}$

How to calculate the value of a decimal raised to a big power without a calculator. I have been using the formula $1-m x+ (m(m-1)/2!) x^2$ but this does not give the correct answer sometimes. Can ...
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2 votes
2 answers
44 views

Evaluating $\sum_{k=a}^{15} \frac{k!}{(k-a)!}$ for $0 \leq a \leq 15$

I am trying to simplify the sum $\sum_{k=a}^{15} \frac{k!}{(k-a)!}$ for a given integer $a \in [0, 15]$. Plugged into WolframAlpha, I see that the expression is equivalent to $\frac{16!}{(a+1) \cdot (...
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0 votes
1 answer
66 views

Evaluation of $~\Delta\left\{\sum_{k=0}^{20}(-1)^{k}\binom{40}{2k}x^{40-2k}y^{2k}\right\}$

$$\Delta f:={\partial^2 f\over\partial x^2}+{\partial^2 f\over\partial y^2}\tag{1}$$ $$f(x,y):= \underbrace{\sum_{k=0}^{20}(-1)^{k}\binom{40}{2k}x^{40-2k}y^{2k}}_{\text{I assume}~(x,y)\neq(0,0)} \tag{...
1 vote
1 answer
58 views

Sum of binomial distribution with increasing trials

I am trying to solve the sum of a specific type of binomial distribution: \begin{align} \sum_{n=0}^{\kappa-s}\binom{s+n-1}{n}\left(1-x\right)^n \end{align} The problem is that the sum affects the ...
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0 votes
2 answers
80 views

if $f(x) = x+2/(1\cdot3)x^3+2\cdot4/(1\cdot3\cdot5)x^5+\cdots$ comment on the value of $f(1/2)$ [closed]

Let $$f(x)= x+\frac2{1\cdot3}x^3+\frac{2\cdot4}{1\cdot3\cdot5}x^5+\frac{2\cdot4\cdot6}{1\cdot3\cdot5\cdot7}x^7+\cdots\quad\forall x\in(0,1)$$ If the value of $f(\frac12)$ is $\dfrac\pi{a\sqrt b}$ (...
0 votes
0 answers
18 views

How to effectively show the following expression equals zero using the binomial theorem [duplicate]

I want to show, using the Binomial Theorem, that $\binom{n}{0} - \binom{n}{1} + \binom{n}{2} + ... + \left ( -1 \right )^{n}\binom{n}{n} = 0$ By using the symmetry of $\binom{n}{0} = \binom{n}{n}$ it ...

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