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

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-6
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0answers
30 views

i want solution of this problem? [on hold]

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2
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0answers
24 views

Tricky divergent binomial expansions?

The binomial expansion of $(a+b)^n$, where $n\notin\mathbb{N}$, is given as $$(a+b)^n=a^n+\frac{n}{1!}a^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^2+\cdots$$ In some situations, we can find the result of a ...
-2
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0answers
26 views

How to calculate the total number of dissimilar terms (terms having different powers in x)… [closed]

The question is that: Calculate the total number of dissimilar terms i.e different powers of x in the binomial expansion of $(1-x+x^{-2}-x^2)^{50}$. I have no idea how to evaluate such type ...
-3
votes
1answer
65 views

Simple Question on Binomial theorems… [closed]

I have tried to solve this question by putting the value of each coefficients but it is really becoming very lengthy.... what i got was this number 10510100501... But how to get this in the required ...
1
vote
2answers
35 views

binomial coefficient where k > n

For solving binomial coefficients we have use from formula $\frac{n!}{k!(n-k)!}$ This formula only works if n > k. What happens if n < k? Is there another formula we need to use?
1
vote
1answer
26 views

Binomial expansion extension to negative powers

I know that: $$\sum_{k=0}^n {n \choose k}a^{n-k}b^k = (a+b)^n$$ But how is this extended to negative powers, for example, I came across the following line of maths, which I struggle to understand: ...
1
vote
1answer
57 views

Show that the sum $\sum_{k = 0}^n 2^k \binom{n}{k}$ is equal to $3^n$

How can I show that the sum $$ \sum_{k = 0}^n 2^k \binom{n}{k}$$ is equal to $3^n$?
0
votes
2answers
42 views

Expand $f(x)=\log(x+ \sqrt{1+x^2})$ into power series and determine convergence intervals

We have $f(x)=\log(x+ \sqrt{1+x^2})$ and we need to expand it into power series, which I suppose is easy because $f'(x)=(1+x^2)^{-1/2}= \sum_{k=0} {-\frac{1}{2} \choose k}x^{2k}$. It follows that ...
6
votes
0answers
55 views

Intuitive explanation of Extended binomial coefficient

We all are familiar with the following formula - $$\dbinom{n}r = \dfrac{n!}{(n-r)! \space r!} \space\space \space ; \space \space n>r$$ This is the binomial formula where $n$ and $r$ are ...
0
votes
1answer
22 views

Generating function for multiset formula

It's said that the generating function for $g(x) = \sum_{d=0}^\infty {d+m-1 \choose m-1} x^d$ is equal to $\frac{1}{(1-x)^m}$. In the proof that I have seen it states that: By the geometric series, ...
2
votes
2answers
50 views

Catalan numbers formula derivation

I'm trying to follow a proof of the Catalan numbers being equal to $\frac{1}{n+1} {2n \choose n}$ from the recurrence relation $C_n = C_0C_{n-1}+C_1C_{n-2}+...+C_{n-2}C_{1}+C_{n-1}C_0$ Now it's seen ...
1
vote
2answers
50 views

Show that a power series is analytic inside its radius of convergence

Let $f(z)=\sum_{k=0}^\infty a_k(z-z_0)^k$ with radius of convergence $R$ then $f$ is analytic on the open disk around $z_0$ with radius $R$. What I was thinking about is an approach based on this ...
3
votes
3answers
39 views

Intuitive proof for a Combinatorial Problem

Given a set $S$ such that $|S|=N$ and $S$ contains exactly $K$ $0$s $(K >0)$ and $N-K$ $1$s, then exactly half of the subsets of $S$ contain an $odd$ number of 1s, $indepedent$ of the value of ...
0
votes
2answers
82 views

Prove for that $(1 + \frac{x}{p})^p < (1 + \frac{x}{q})^q$ [without Bernoulli inequality or integrals] [duplicate]

Prove for $x > 0$ and $0 < p < q$ that $$(1 + \frac{x}{p})^p < (1 + \frac{x}{q})^q$$ I think that binomial theorem might be of use in this exercise, but I'm not sure. I haven't been able ...
0
votes
0answers
41 views

Calculating summations concerning binominal coefficients

Since $$\sum_{j=0}^{m}\binom{k}{j}\binom{k}{m-j}=\binom{2k}{m},$$ what is the result of $$\sum_{j=0}^{m}\binom{k}{j}\binom{k}{m-j}(\frac{1}{3})^{j}=?$$ Here $i, j, k, m$ are integers.
4
votes
1answer
114 views

Find the coefficient of $x^{19}$ in the expression $(x+1)(x+2)(x+3)\cdots (x+400)$

Find the coefficient of $x^{19}$ in the expression $(x+1)(x+2)(x+3)\cdots (x+400)$ I have no clue how to start. Any kind of help will be appreciated.
3
votes
3answers
59 views

Finding the coefficient of expansion

Question: Find the coefficient of $x^{11}$ in the expansion of:$$(1+x^2)^4(1+x^3)^7(1+x^4)^{12}$$ The traditional way of doing this, as far as I know, is to first find the coefficient of each term ...
0
votes
1answer
23 views

How do I work out the validity for a Maclaurin (power) series?

I cannot find the answer to this anywhere so I have decided to make a question. Given a Maclaurin series for a function, how can I quickly work out what the validity is for it? For example, ...
8
votes
1answer
75 views

Divisibilty of $1^{101} + 2^{101} + 3^{101}+ 4^{101}+\cdots+2016^{101}$ [duplicate]

$1^{101} + 2^{101} + 3^{101}+ 4^{101}+\cdots+2016^{101}$ is divisible by which of the following? $(A)$ $2014$ $(B)$ $2015$ $(C)$ $2016$ $(D)$ $2017$ Could someone share the approach to deal with ...
1
vote
1answer
27 views

Can this be done without substitution of values?

If $${ \left( 1-{ x }^{ 3 } \right) }^{ n }=\sum _{ r=0 }^{ n }{ { a }_{ r }{ x }^{ r }{ \left( 1-x \right) }^{ 3n-2r } } $$ then find $a_r$. My first attempt: I wrote the above equation as: ...
1
vote
1answer
35 views

Remainder when $2005^{2002} + 2002^{2005}$ is divided by $2003$

Find the remainder when $2005^{2002} + 2002^{2005}$ is divided by $2003$ ? Is there any better approach to this rather than using binomial theorem?
0
votes
1answer
38 views

What is the closed form representation of the sum of the first $\text{int}(n/2)$ terms of binomial expansion $(f+(1-f))^n$?

Say that we have this polynomial $(f + (1-f))^n$ where $f$ and $n$ are some positive real numbers, except that $f$ is a constant, but $n$ is a variable. That term can be expanded using the binomial ...
1
vote
1answer
27 views

Binomial Expansion - Finding the term independent of n.

The coefficient of $x^2$ in the expansion of $\left(1 + \frac x5\right)^n$, where $n$ is a positive integer, is $\frac 35$ . $(i)$ Find the value of $n$. $(ii)$ Using this value of $n$, ...
0
votes
1answer
23 views

How to simplify following binomial expansion?

$\dbinom{2n+1}{n+1}+\dbinom{2n+1}{n+2}+....+\dbinom{2n+1}{2n }+\dbinom{2n+1}{2n+1}$? my textbook's answer is $2^{2n}$.
0
votes
2answers
30 views

Prove by induction: the coefficients of (a+b) to the power of n are the same if turned into a number as 11 to the power of n

Proof by induction that the coefficients of $(a+b)^n$ in order, if place as a number, the first coefficient being having the biggest place value, and each number lowers in place value, are equal to ...
3
votes
5answers
90 views

Knowing binomial expansion of $(1+kx)^n$ shortcut for $(a+kx)^n$

If I've calculated the binomial expansion of $(1+kx)^n$ (where $k$ and $n$ are known) up to some term, is there then a shortcut for calculating $(a+kx)^n$? Example: Given that the binomial ...
1
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1answer
37 views

Inverse Euler theorem to calculate $\binom{n}{r}$

How can we use Inverse Euler theorem or properties to calculate the binominal coefficients or say $\binom{n}{r}$? What is the algorithm for this ? An example for the same will be greatly ...
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1answer
29 views

approximation of binomial coefficient by exponentiation

Show that inequality holds for every n, k. $$\ { n \choose k} \leq \frac{n^k}{k!}. ( \, 1-\frac {k}{2n} ) \,^{k-1} $$ I used Stirling's formula but I stucked, which is $$\ { n \choose k} = ...
1
vote
1answer
50 views

Problem with binomial coefficients?

I am trying to find the sum of $$\sum_{x=0}^{n-2}\left (\frac{1}{x+1}{2x \choose x} \cdot \frac{1}{n-x-1}{2n-2x-4 \choose n-x-2}\right)\;.$$ I am told the answer is $$\frac{1}{n}{2n-2 \choose n-1}$$ ...
-2
votes
1answer
59 views

Value of sum of binomials: $P = \binom{N}{0}-\binom{N}{1}+\binom{N}{2}-\binom{N}{3}+ \dotsb + (-1)^N\binom{N}{N}$ [duplicate]

$P = \binom{N}{0}-\binom{N}{1}+\binom{N}{2}-\binom{N}{3}+ \dotsb + (-1)^N\binom{N}{N}$ I can calculate the value of this equation manually, but there any direct formula for calculating the value of ...
4
votes
2answers
109 views

Solving ${98 \choose 30}+2{97 \choose 30}+3 {96 \choose 30}+…+68{31 \choose 30}={100 \choose q}$ for $q$

${98 \choose 30}+2{97 \choose 30}+3 {96 \choose 30}+...+68{31 \choose 30}={100 \choose q}$ Find the value of $q$? Could someone give me hint as how to solve this question?
0
votes
1answer
30 views

Binomial inequality problem ${k+n-1 \choose k}\times{k+n+1 \choose k} \leq{k+n \choose k}^2$

Can anyone help we with this problem: Let $a_n={k+n \choose k} $ Prove that $a_{k-1}a_{k+1}\leq a_k^2 $($\forall k$) My first idea was using mathematical induction to proof that for every k element of ...
0
votes
3answers
26 views

show equality - binomial formula, taylor?

I am trying to show that this is true using the binomial formula or some taylor expansion: $\frac{1}{(1+\epsilon \sum\limits_{n=0}^\infty Z_n(t) \epsilon^n)^2} = 1 - 2Z_0\epsilon + ...
0
votes
1answer
21 views

How find the value of $(a_0-a_2+a_4-\ldots)^2+(a_1-a_3+ \ldots)^2$ using $(1+x)^n=a_0+a_1x+a_2x^2+\ldots+a_nx^n$?

Q) $(1+x)^n=a_0+a_1x+a_2x^2+\ldots+a_nx^n$ then $(a_0-a_2+a_4-\ldots)^2+(a_1-a_3+ \ldots)^2$ is equals to 1. 12. 0 (zero)3. $2^{n-1}$4. $2^n$ Answer: (4) well this time i am rocked by this ...
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votes
2answers
31 views

Binomial theroem problem

I have a problem and i think that will be solved by binomial theorem I am not sure: $20^n+16^n-3^n-1$ get $n$ to make that expression divisible by $323$.
4
votes
4answers
212 views

How is the binomial expansion of the vectors?

I'm trying to find out if there is an attempt to define binomial expansion of vectors. i.e $$(\overrightarrow a + \overrightarrow b)^n = ?? $$ I tried to google around this (e.g : binomial expansion ...
0
votes
1answer
24 views

Why does $\sum_{k\geq0}\binom{-r}{k}p^r(p-1)^k=p^r(1+p-1)^{-r}$?

For any positive real number $r$, it is clear that $\binom{-r}{k}(-1)^k\geq0$ for all positive integer $k$. The general binomial theorem then implies ...
4
votes
1answer
50 views

Using binomial theorem to prove an identity

I'm asked to prove the identity $$n(n+1)2^{n-2} = \sum_{k=1}^{n}k^2\binom{n}{k}$$ using the binomial theorem. What I've been able to come up with so far is letting $$f(x) = (x+1)^n = ...
1
vote
1answer
38 views

Help with series definition, binomial theorem?

If a stock goes up by a factor $u$ with probability $p$ and down by a factor $d$ with probability $1-p$, find the expected value of the stock after $n$ periods. Assume the periods are independent. I ...
2
votes
2answers
48 views

If $(1+ax+bx^{2})^{10} = 1-30x+410x^{2}+…$ find the value of a and b then find the coefficient of $x^{19}$ in this expansion.

If: $$(1+ax+bx^{2})^{10} = 1-30x+410x^{2}+...$$ find the value of a and b then find the coefficient of $x^{19}$ in this expansion. I found $a=-3$ and $b=\frac{1}{2}$ by writing the trinomial as a ...
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votes
7answers
93 views

Determine the constant term in the development of $\left(x+\frac1x\right)^6$ [closed]

Please help me solve this.. I think it's done with binomial theorem? Determine the constant term in the development of $\left(x+\frac1x\right)^6$
2
votes
2answers
89 views

Binomial Theorem Sum [closed]

How would I find the sum below? $$ \sum_{k=0}^{n} {\frac{1}{k+1} {n \choose k}}=?$$ I need help starting on the problem. Would I first need to utilize the Binomial Theorem proof to get started on ...
3
votes
2answers
152 views

Evaluating a “binomial-like” sum

I suspect there is a way to do the following sum by hand, but I'm having some trouble: $$\sum_{x=0}^{n} x^{2} {n \choose x} p^{x}(1-p)^{n-x}$$ There are a couple questions like this, but for general ...
3
votes
1answer
24 views

Prove $\displaystyle\sum_{r=1}^{n-2} r.\binom{n-r}{2}=\binom{n+1}{4}$

How to prove $\displaystyle\sum_{r=1}^{n-2} r.\binom{n-r}{2}=\binom{n+1}{4}$? I tried writing it as an AGP as following: $$\displaystyle\sum_{r=1}^{n-2} r.\binom{n-r}{2} = \textrm{coefficient of } ...
2
votes
2answers
60 views

find the value of $\int_0^1(C(-y-1)\cdot\sum_{k=1}^{2016}\frac{1}{y+k})dy$

The problem: find the value of $\int_0^1 (C(-y-1)\cdot\sum_{k=1}^{2016}\frac{1}{y+k}) dy$, where $C(\alpha$) is the coefficient of $x^{2016}$ in the Maclaurin series for $(1+x)^\alpha$ What I ...
1
vote
1answer
31 views

Prove that $n\cdot(x+1)^{n-1} = \sum_{k=0}^{n}k\cdot\binom{n}{k}\cdot x^{k-1}$

I have to prove that $n\cdot(x+1)^{n-1} = \sum_{k=0}^{n}k\cdot\binom{n}{k}\cdot x^{k-1}$. I know how to prove it by expand the binomial theorem of $(x+1)^{n}$ and then derive it. But, I have to ...
0
votes
1answer
47 views

find $\frac{3}{6}+\frac{3\cdot5}{6\cdot9}+\frac{3\cdot5\cdot7}{6\cdot9\cdot12}+\cdots$ [duplicate]

find $\frac{3}{6}+\frac{3\cdot5}{6\cdot9}+\frac{3\cdot5\cdot7}{6\cdot9\cdot12}+\cdots$ I had $(1-x)^{-\frac{p}{q}}$ in mind. ...
1
vote
2answers
52 views

Prove $\sum_{0}^{n}(-1)^i\binom{n}{i} = 0$ [closed]

I have to prove that $\sum_{0}^{n}(-1)^i\binom{n}{i} = 0$, I know i have to use the binomial theorem,But i dont know how. Thanks.
1
vote
1answer
11 views

Binomial expansion and convergence

May I ask for some help on this?
2
votes
0answers
40 views

Binomial Theorem Proof by Induction

Did i prove the Binomial Theorem correctly? I got a feeling I did, but need another set of eyes to look over my work. Not really much of a question, sorry.