Questions tagged [binomial-theorem]

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

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Summation of Binomial Coefficient containing negative r

In my book it is given: $$\sum_{r=0}^nr(^nC_r)=n\cdot2^{n-1}$$ But if I prove it this way $$\implies\sum_{r=0}^nr(^nC_r)$$ $$\implies \sum_{r=0}^nr\cdot\left(\frac{n}{r}\right)\space ^{n-1}C_{r-1}$$ $$...
ADITYA DAS's user avatar
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1 answer
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Prove that the coefficient of equidistant terms in the expansion of $(1+x+x^{2})^{2}$ are equal.

The Actual Question: If $$(1+x+x^{2})^{n} = a_0+a_1x+a_2x^{2}+\cdots+(a_{2n})x^{2n}$$ then $$(a_0+a_1+\cdots a_n)^{3}-(a_n +a_{n+1}+\cdots a_{2n})^{3}$$ is equal to? Experience says that equidistant ...
Ayush Naman's user avatar
1 vote
2 answers
125 views

How do you prove this statement: $\frac{5}{1\cdot2\cdot3}+\frac{7}{3\cdot4\cdot5}+\frac{9}{5\cdot6\cdot7}+\cdots =-1+3\ln\left(2\right)$

How can prove this statement with the help of this property :$\ln\left(1+x\right)=\sum_{n=1}^{\infty}\frac{\left(-1\right)^{\left(n+1\right)}x^{n}}{n}$ What i did was, i found the general term i.e $t_{...
3b1b aimer's user avatar
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33 views

Recurrence relation for integer sequences raised to the power of $n$

An interesting pattern can be observed when considering a sequence of positive integers raised to some power $n$. When a sequence of continuous integers $i$ (of length >=n) is raised to the power ...
Edison Medison's user avatar
2 votes
0 answers
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Application of the Principle of Inclusion/Exclusion and the Binomial Theorem in Combinatorial Proofs [closed]

Consider a set $Z=X \cup Y$, where $X=\left\{x_1, \ldots, x_n\right\}$ is a set of blue elements and $Y=$ $\left\{y_1, \ldots, y_m\right\}$ is a set of red elements. (a) How many subsets of $Z$ ...
Allison's user avatar
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How to prove this inequality relating to Bernoulli trials? [duplicate]

The inequality is verbally intuitive, however an algebraic approach seems to be not as easy: Consider $n$ Bernoulli trials, where for $i=1,2,\ldots,n$, the $i$ th trial has probability $p_i$ of ...
excitedGoose's user avatar
-4 votes
1 answer
106 views

Problem with kids playing a game on a grid with lattice points [closed]

A group of 252 kids play a game. They draw a grid on the ground consisting of lattice points and have to get from point $(0,0)$ to $(5,5)$ (the kids can only take unit length steps up or to the right)....
math.enthusiast9's user avatar
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How can I prove this approximation of a binomial distribution?

The exercise demands to show that the value of the maximum of the binomial distribution $b(k;n,p)$ is approximately $\frac{1}{\sqrt{2\pi npq}}$ , apparently there is some relation with Stirling's ...
excitedGoose's user avatar
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Proving $\sum_{i = b}^{a+b-1}\binom{i-1}{b-1}\cdot i\cdot 2^{b+a-i} = 2\cdot b\cdot \sum_{i = b+1}^{a+b} \binom{i-1}{b} \cdot 2^{b+a-i}$

Dear Math stackexchange community, I recently stumbled across the following binomial identity when working on a combinatoric project: $$\sum_{i = b}^{a+b-1}\binom{i-1}{b-1}\cdot i\cdot 2^{b+a-i} = 2\...
potenzenpaul's user avatar
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How to prove this binomial lower bound?

I have been stuck in this problem for a while and no prolific idea comes up to my mind, could anyone indicate a path to its solution? $$\begin{pmatrix}n\\k\end{pmatrix}\geq\frac{n^k}{4k!}$$ for $k \...
excitedGoose's user avatar
-1 votes
1 answer
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How this binomial upper bound can be proven? [duplicate]

Could anyone shed some light in how this inequality can be proven ? (inductive approach is suggested, but if there is another method it would be great too) $$\begin{pmatrix}n\\k\end{pmatrix}\leq\frac{...
excitedGoose's user avatar
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2 answers
48 views

How to solve this binomial summation? [duplicate]

I was solving my problem sheet and came upon this problem. original problem I deduced the problem to this form, but I can't solve further. Can you please provide some help. $$\sum_{r=0}^{12}\binom{13}{...
satyam singh's user avatar
2 votes
3 answers
104 views

When $(x+1)^{m}$ is multiplied out, find the value of $m.$

When $(x+1)^{m}$ is multiplied out, we get the number of coefficients that are not divisible by $2,3$ and $5$ are $64,16$ and $8$ respectively. Find the value of $m.$ I have an idea of trying Lucas ...
Debrogli's user avatar
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Can we use Binomial Approximations when evaluating limits?

I came across this question, Compute $$L=\lim\limits_{x\to 0}{\frac{\sqrt[3]{1+\sin^2 x} \hspace{2mm}-\sqrt[4]{1-2 \tan x}}{\sin x + \tan^2 x}}$$ Can I use Binomial Approximations here? As $x\to 0 \...
Jesko's user avatar
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AB is a chord of length 2ka of a circle of radius a. The tangents to the circle at A and B meet in C if k^7 is negligible calculate the area of ABC

AB is a chord of length 2ka of a circle of radius a. The tangents to the circle at A and B meet in C. Show that, if k is so small compared with unity that $k^7$ is negligible, the area of the triangle ...
Skwaches's user avatar
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1 answer
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How to Prove the Binomial Series by Differentiation? [closed]

(a) Let $g(x) = \sum_{n=0}^\infty \binom{k}{n} x^n$. Differentiate this series to show that $$g'(x) = \frac{kg(x)}{1+x}, \quad -1 < x < 1.$$ (b) Let $h(x) = (1+x)^{-k} g(x)$ and show that $h'(x) ...
Dragic98's user avatar
1 vote
1 answer
44 views

$ \sum_{k=0}^{n}\mathrm{C}_{2k}^{n+k} {(-4)}^{k} $

Let $ S_{n}=\sum_{k=0}^{n}\mathrm{C}_{2k}^{n+k} {(-4)}^{k} $ then prove that $ S_{n+1}+2S_{n}+S_{n-1} = 0 $ (where n is a natural number greater than 2 and $ \mathrm{C}_{k}^{n} = \binom{n}{k} $) so ...
user483801's user avatar
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0 answers
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A property of product of sequence being small

Let $a \in (0,1)$ and a monotonically (strictly) decreasing sequence of non-negative numbers $t_n \to 0$ be given. Define $$b_n = t_n(1+a) + a.$$ How can I show that for every $\epsilon >0$, there ...
C_Al's user avatar
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2 votes
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An application of binomial theorem

I came across this question in Pure Mathematics 2 in The Binomial Theorem section. The first part is simple and goes like this: If a clock with a seconds pendulum registers x seconds too few per day, ...
Skwaches's user avatar
2 votes
2 answers
64 views

compute $\sum_{k=0}^{n} \frac{{(-1)^k}}{k+1}{n \choose k}$.

Find $\sum_{k=0}^{n} \frac{{(-1)^k}}{k+1}{n \choose k}$ as a function of n. I have done it in the following way: Notice first that $\sum_{k=0}^{n} \frac{{(-1)^k}}{k+1}{n \choose k} = \sum_{k=0}^{\...
ofirsasoni's user avatar
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1 answer
38 views

Binomial Theorem Coefficients derivation

By the Binomial Theorem, when $y<1$: $(1+x)^y=1+yx+\dfrac{y(y-1)}{2!}x^2+\dfrac{y(y-1)(y-2)}{3!}...$ The coefficient of $y$ is: $x+\dfrac{(-1)}{2!}x^2+\dfrac{(-1)(-2)}{3!}x^3+\dfrac{(-1)(-2)(-3)}{4!...
ronald christenkkson's user avatar
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0 answers
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Cumulative Distribution Function (CDF) of a Binomial Distribution

I was solving the following question: Find the CDF of a Binomial Distribution. For some reason, I missed the fact that the CDF of a Binomial Distribution is simply $$F(x) = \sum_{i=0}^x \binom{n}{i}p^...
Sifiso Rimana's user avatar
1 vote
1 answer
36 views

Arithmetic sequence in coefficients of $(x+y)^n$

For which $n$ the second and third and fourth term's coefficients of binomial expansion's $(x+y)^n$ makes arithmetic sequence?
Ali's user avatar
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3 votes
2 answers
129 views

Proof $\displaystyle \sum_{k=0}^n \binom{2n}{2k} = 2^{2n-1}$ [duplicate]

Can someone help me prove the equation: $$\sum_{k=0}^n \binom{2n}{2k} = 2^{2n-1}$$ I know that via binomial theorem for even k: $$2^{2n} = (1+1)^{2n} = \sum_{k=0}^n \binom{2n}{k}$$ and for odd k: $$0 =...
Mar's user avatar
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1 answer
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Find $n$ when the coefficients of the $16^{th}$ and $26^{th}$ terms of $(1+x)^n$ are equal.

Find $n$ when the coefficients of the $16^{th}$ and $26^{th}$ terms of $(1+x)^n$ are equal. $16^{th}$ term coefficient: $^nC_{15}= \space ^nC_{n-15}$ $26^{th}$ term coefficient: $^nC_{25}= \space ^...
ronald christenkkson's user avatar
3 votes
1 answer
63 views

What is the total number of words of length $500$ on $\{a,b\}$ such that the letter $"a"$ appears more than $"b"$ ( without Brute force)?

The question : What is the total number of words of length $500$ on $\{a,b\}$ such that the letter "$a$" appears more than "$b$"? $(*)$ We know that the total number of words is $ ...
DanielMa's user avatar
  • 149
0 votes
1 answer
61 views

high order derivative of product

Let $f\in\mathcal{C}^\infty(\mathbb{R})$, what is the form of $$ \frac{d^n}{dx^n}\left(\frac{f(x)}{x}\right) $$ for any $n\in\mathbb{N}$? I need to pull out $\frac{d^n}{dx^n}f(x)$ if possible. Thank ...
Giulio Binosi's user avatar
2 votes
1 answer
76 views

Evaluating a binomial sum with a probabilistic interpretation

Evaluate the following double summation over all acceptable values of $m$ and $k$: $$S=\sum_{0 \leq k < m \leq n+1} \binom {n+1}{m}\binom {n}{k}$$ This sum arose in a probability question a friend ...
Cognoscenti's user avatar
0 votes
0 answers
21 views

Sum of products from $k=1$ to $m$ ( Context: Burton's proof of the binomial theorem)

This is an elementary question asked by the beginner at the frontdoor of higher mathematics. In Burton's Elementary Number Theory ( seventh edition, page 9) I find the following equality written ...
Vince Vickler's user avatar
1 vote
0 answers
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Is Wikipedia's inductive proof of the Binomial Theorem acceptable?

I thought Wikipedia had a really elegant proof by induction of the binomial theorem. However, I wondered if the reasoning made at the line marked by (*) below is acceptable. When $j = 0$, $k = n + 1$, ...
ten_to_tenth's user avatar
3 votes
4 answers
111 views

Proving $\sum_{i=0}^n (-1)^i\binom{n}{i}\binom{m+i}{m}=(-1)^n\binom{m}{m-n}$

I am trying to prove the following binomial identity: $$\sum_{i=0}^n (-1)^i\binom{n}{i}\binom{m+i}{m}=(-1)^n\binom{m}{m-n}$$ My idea was to use the identity $$\binom{m}{m-n}=\binom{m}{n}=\sum_{i=0}^n(-...
Hjlmath's user avatar
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1 vote
1 answer
52 views

In $\sum_{n=1}^{\left\lfloor \delta/2 \right\rfloor}$, what does ${\left\lfloor \delta/2 \right\rfloor}$ mean? [closed]

What does the symbol ${\left\lfloor \delta/2 \right\rfloor}$ mean here? $$\sum_{n=1}^{\left\lfloor \delta/2 \right\rfloor}$$ Any reference on understanding its usage in binomial theorem would be ...
NovoGrav's user avatar
9 votes
1 answer
243 views

Another Sophomore's Dream: $\int_{-\infty}^{\infty}\binom{n}{x}dx=\sum_{i=0}^{n} \binom{n}{i}$

I found an identity $$\int_{-\infty}^{\infty}\binom{n}{x}dx=\sum_{i=0}^{n} \binom{n}{i}$$ where LHS can be calculated by the Reflection relation and Dirichlet integral. The result is $2^n$, which is ...
user32688's user avatar
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1 vote
1 answer
51 views

Absolute Value Condition in Generalized Binomial Theorem

What is the coefficient of $\frac{y^3}{x^8}$ in $(x+y)^{-5}$? when $|\frac{y}{x}|<1$ (EAMCET 2020) I am using the generalized binomial theorem $$(x+y)^r=\sum_{k=0}^{\infty}{r \choose k} x^{r-k}y^...
Starlight's user avatar
  • 1,680
1 vote
3 answers
131 views

Evaluate the sum $\sum_{k=0}^n \frac{\binom n k}{(k+1)(k+3)} $

How to sum the sequence $$\sum_{k=0}^n \frac{\binom n k}{(k+1)(k+3)} $$ where $n \choose k$ are the usual binomial coefficients in expansion of $(1+x)^n$. I know how to sum such sequence when the ...
Eisenstein's user avatar
-1 votes
2 answers
37 views

Prove $(k+1)\binom{n}{k+1}+k\binom{n}{k}=n\binom{n}{k}$ for integers $0\le k\le n$

Prove $(k+1)\binom{n}{k+1}+k\binom{n}{k}=n\binom{n}{k}$ for integers $0\le k\le n$ I need help, I've been trying to factor all day and can't figure it out.
Hanna Vink's user avatar
1 vote
1 answer
65 views

How many $n$-digit numbers are there with no or even number of $1$s in them if only digits $1$, $2$, $3$ and $4$ are allowed?

You are given an unlimited supply of each of the digits 1,2,3 or 4. Using only these four digits, you construct n digit numbers. Such n digit numbers will be called LEGITIMATE if it contains the digit ...
Daksh's user avatar
  • 97
1 vote
0 answers
27 views

binomial approximation doubt

suppose question is $$\frac{1}{(1+x)^3}$$ where $x<<1$ so answer will be $1-3x$ by binomial approximation. But if we do binomial approximation in denominator itself,answer will be $\frac{1}{(1+...
Harshita Jain's user avatar
0 votes
1 answer
49 views

Can we solve this only by using bijection?

The question is : The number of possible outcomes in a throw of n ordinary dice in which at least once of the dice shows an odd number are: Now, we can simply apply bijection principle , and ...
Adhway's user avatar
  • 149
1 vote
0 answers
68 views

Confusion regarding the nth term in a binomial expansion (positive integral index)

The expansion for a binomial is - $(a+b)^n=\sum_{k=0}^n{n\choose k}a^{n-k}b^{k} $ But it could also be this $(a+b)^n=\sum_{k=0}^n{n\choose k}a^kb^{n-k}$ Both of the expansions are correct because the ...
IshA's user avatar
  • 24
-2 votes
1 answer
52 views

Binomial coefficient identity with alternating summands [closed]

Is there a nice closed form for $\sum_{k=0}^{l} (-1)^k \binom{n}{k} \binom{n-k}{l-k}.$ I feel like there must be but I cannot find it.
wecanfibonacciit's user avatar
1 vote
4 answers
132 views

The coefficient of $x^{101}$ in the expansion of the expression $(5+x)^{500}+x(5+x)^{499}....+x^{500}$ is

The coefficient of $x^{101}$ in the expansion of the expression $$(5+x)^{500}+ x(5+x)^{499}+x^2(5+x)^{498}+\cdots+x^{500}$$ is? My attempt:- The coefficient of $x^{101}$ in the first term is $C(500,...
math and physics forever's user avatar
2 votes
3 answers
136 views

Exploring the Limiting Behavior of $(n + 1)^n$ in Base $ n $ as $ n $ Approaches Infinity

Hello Math Stack Exchange Community, I've been investigating an interesting pattern in the polynomial expansion of $(n + 1)^n$ when expressed in base $ n $, specifically the emergence of a consistent ...
NeonNarwhal's user avatar
0 votes
0 answers
39 views

Two forms of $B_k$, if $(1+x-x^2)^n=\sum_{k=0}^{2n} B_k x^k$

Earlier [ https://math.stackexchange.com/questions/4841673/finding-b-k-if-1x-x2n-sum-k-02n-b-k-xk ] it has been shown that $$B_k=\sum_{j=0}^{k} (-1)^{k-j} {n \choose j}{j \choose k-j}.......(1)$$ And ...
Z Ahmed's user avatar
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0 votes
1 answer
58 views

Finding $B_k$, if $ (1+x-x^2)^n=\sum_{k=0}^{2n} B_k x^k$

For $0\le k \le n$, we can write $$B_k= \text{Coef. of}~ x^k ~\text{in}~(1+x-x^2)^n$$ $$\implies B_k=\text{Coef. of}~ x^k ~\text{in}~\sum_{j=0}^{n} {n \choose j} x^j(1-x)^j$$ $$=\sum_{j=0}^{n} {n \...
Z Ahmed's user avatar
  • 43.2k
0 votes
0 answers
77 views

Sum of a binomial expansion

I came across this question: Sum the series $${6n \choose 0} + {6n \choose 3} + {6n \choose 6} + \cdots + {6n \choose 3n}$$ I tried putting different values of $n$ and tried to find a pattern in the ...
Sourav Bhattacharyya's user avatar
1 vote
0 answers
79 views

Proving an equivalent of Binomial Theorem

Prove that: $$(\alpha+\beta)^n = \sum_{k=0}^{\infty} \binom{n}{k} (\alpha + \beta - 1)^{k} \quad \text{when } |\alpha+\beta-1|<1$$ I got the latter expression from Wolfram Alpha in Series ...
skyfall's user avatar
  • 181
2 votes
1 answer
90 views

Equivalence of two binomial sums

Earlier (Coefficient of terms in expansions) when $f(x)=(1+x+x^2)^n=\sum_{k=0}^{2n} A_k x^k$, $A_k$ has been obtained as $$A_k=\sum_{j=0}^{k} {n\choose j} {j \choose k-j}.......(1)$$ Alternatively, we ...
Z Ahmed's user avatar
  • 43.2k
2 votes
2 answers
118 views

Binomial Theorem proof using three terms

If $$(6-12x+12x^2)^n =\sum\limits_{r=0}^{2n}a_{r}x^r$$ then prove that $$a_{r} = (-1)^r3^n2^r\left[{2n \choose r} + {n \choose 1}{(2n - 2) \choose r} + {n \choose 2}{(2n-4) \choose r} + \cdots \right]$...
Sourav Bhattacharyya's user avatar
2 votes
1 answer
106 views

Finding $ \sum_{k=0}^n (-1)^k A_k {n \choose k}, ~\text{if}~ (1+x+x^2)^n=\sum_{k=0}^{2n} A_k x^k$

Finding $S=\sum_{k=0}^n (-1)^k A_k{n\choose k}$, if $(1+x+x^2)^n=\sum_{k=0}^{2n} A_k x^k.$ \begin{align} (1+x+x^2)^n &= A_0+A_1x+A_2x^2+\dots+A_nx^n+\dots+A_{2n}x^{2n} \\ (1-1/x)^n &= {n \...
Z Ahmed's user avatar
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