Questions tagged [binomial-theorem]
For questions related to the binomial theorem, which describes the algebraic expansion of powers of binomials.
4
votes
1answer
59 views
Combinatorial Analysis - Specific problem
I am having difficulty modeling a combinatorial analysis on a particular problem, I wanted to isolate some generic form to count how many valid arrangements exist in a given problem, can anyone help ...
1
vote
2answers
40 views
Prove that $(n+2)^n < (n+1)^{n+1}$ for all $n \in \Bbb N$
Is there a simple(ish) way of proving this? I got to this step when I was trying to show that the sequence $a_n = (1 + \frac{1}{n})^n$ is increasing for all $n \in \Bbb N$. It came up after I expanded ...
0
votes
3answers
28 views
Find the values of $a$ and $b$ when the binomial expansion of $\frac{1}{(1+ax)^b}+\frac{1}{(1+bx)^a} = 2-6x+15x^2$
Find the values of $a$ and $b$ when the binomial expansion of $$\frac{1}{(1+ax)^b}+\frac{1}{(1+bx)^a} = 2-6x+15x^2$$
So I set up the two equations:
$$(-b)(ax)+(-a)(bx)=-6x$$ and $$\frac{(-b)(-b-1)(ax)...
-1
votes
2answers
35 views
Binomial expansion of $\frac{1}{1+x+x^2}$ up to the first three terms [closed]
Binomial expansion of $\frac{1}{1+x+x^2}$ up to the first three terms
I am unsure where to start with this as i cannot put it into partial fractions, so don't really have an idea on where to start and ...
0
votes
3answers
19 views
Finding the term independent of $x$ from $2$ expansions.
I was asked to find the first $3$ terms of the expansion $\left(3-\frac1{9x}\right)^5$ and was further asked to find the term independent of x in the expansion of $\left(3-\frac1{9x}\right)^5(2+9x)^2$....
0
votes
1answer
26 views
Find the binomial expansion of the function up to the first 3 non-zero terms $\sqrt[3]\frac{1+2x}{1-x}$
Find the binomial expansion of the function up to the first 3 non-zero terms
$$\sqrt[3]\frac{1+2x}{1-x}$$
The function can be broken into $(1+2x)^{\frac{1}{3}}$ and $(1-x)^{\frac{-1}{3}}$
where
$(1+...
0
votes
1answer
18 views
Explanation In the orderings involving $2$ kinds of elements
A population of $a$ indistinguishable alphas and $b$ indistinguishable betas can be arranged in $\binom{a+b}{a}=\binom{a+b}{b}$ distingishable orders.
Any permutations among the alphas, or among ...
-1
votes
2answers
49 views
Rule of 72 binomial
Rule of 72?
So you have to work out the time it takes for an investment to double. With interest rate = x
So I get:
(1+x)^n = 2
Then I get how the 72 rule is formed through solving for n, giving n ...
2
votes
1answer
39 views
Probability and Statistics: Over-booking flights
I am finding difficulty trying to answer the second part of this question. The answer is one this website but I do now get how they worked it out. https://nrich.maths.org/4932
I also know that ...
0
votes
2answers
26 views
Using the Binomial Theorem to expand the magnitude of the difference of two vectors
I have the following expression that I need to expand using the Binomial Theorem: $$\frac{1}{\mid\vec{r}-\vec{d}\mid}$$
Now the Binomial Theorem is the following:
$$(x+y)^r = \sum^{\infty}_{k=0}\binom{...
2
votes
3answers
54 views
prove that ${2n \choose n} \le 4^{n}$
i need to prove that ${2n \choose n} \le 4^{n}$ for all $ n \in N $.
I tried to prove it that way:
$$ 4^{n} = (1+1)^{2n} = \sum_{i=0}^{2n} {2n \choose i}1^{2n-i}1^{i} = \sum_{i=0}^{2n} {2n \choose i} ...
0
votes
4answers
58 views
How to prove divisibility of a number using the binomial expansion? [closed]
I have the following problem:
Prove that $6^n-1$ is always divisible by 5 using the binomial expansion of $(5+1)^n$.
How can I do this? I don't know how to begin, as I don't see how the binomial ...
-1
votes
2answers
30 views
Finding the value of a double summation [closed]
The question says to find the value of
$$\sum ^{n} _ {r=1} \sum ^ {r-1} _ {p=0} \binom{n}{r} \cdot \binom {r}{p} \cdot 2^p$$
I have no idea how to deal with the double summation, I just know that ...
0
votes
1answer
78 views
Prove that $(x+y)^p=x^p+y^p \mod p$ where $p$ is a prime.
Let $p$ be a prime, we want to prove that $(x+y)^p=x^p+y^p \mod p$.
The left handside looks like something we can throw the binomial theorem at, also in the previous proof the book asked to prove that ...
1
vote
1answer
28 views
Bacteria that doubles every time step with a probability of p or dies. What is expected number of bacteria after n time steps?
As stated in the problem. Given a bacterium that at every time step $t$, either divides with a probability $p$ or dies off. What is the expected number of bacteria after $n$ time steps?
Does this use ...
1
vote
2answers
55 views
Prove $3^{n−2} n (n − 1) = \sum_{k=2}^{n} \binom{n}{k} (k)(k − 1)2^{k−2}$ for $n\ge3$
The question asks for a combinatorial proof only.
In my attempt, I rewrote $3^{n-2}$ as $(1+2)^{n-2}$. Then using binomial theorem, i was able to get $(1+2)^{n-2} = \sum_{k=2}^{n} \binom{n}{k} 2^k$....
1
vote
1answer
38 views
Power Series / Taylor Expansion of $\frac{1}{\left(1-t^2\right)^{\frac{1}{2}}}$
The question:
$\left(\frac{1+t}{1-t}\right)^{\frac{1}{2}}\:=\:\sum _{n=0}^{\infty }\:a_nt^n$
The question asks for what is $a_n$
These are the steps I've done:
$\left(\frac{1+t}{1-t}\right)^{\frac{...
1
vote
0answers
32 views
Characteristic Function and Generalized Binomial Theorem
could you please help me on the following problem?
I have the characteristic functions of two random variables $X$ and $Y$, denoted $\phi_X(u)$ and $\phi_Y(u)$, such that:
\begin{equation*}
\...
0
votes
1answer
65 views
Approximate $\sqrt{7}$ using binomial theorem [duplicate]
How does one deduce the approximation of $\sqrt{7}$ to be $\frac{10837}{4096}$ by taking $x = \frac{1}{64}$ in the expansion of $\sqrt{1-x}$?
How should you approach such a question? I assume the ...
0
votes
0answers
19 views
Binomial expansion; first four terms when $x < a$ or $x > a$
Using the Binomial Theorem, how do you find the first four terms of the expansion of$$(x+3)^{-4}$$when $x<3$ and when $x>3$ ?
2
votes
1answer
54 views
O level Expansion problem. [duplicate]
yep, title sucks..
Q: The expansion of (1+px+qx2)8 = 1+8x+52x2+kx3.
Find the values of p, q and k.
I found the values of $p$ and $q$ and they are:
p = 1
q = 3
But I am unable to find the value ...
2
votes
1answer
38 views
Prove -1 Transformation (A binomial identity):$\binom{-p}{q}=\binom{p+q-1}{q}(-1)^q$
Prove the following binomial identity :
$$\begin{align*}
\binom{-p}{q}=\binom{p+q-1}{q}(-1)^q
\end{align*}
$$
I played with $\binom{-p}{q}$ for a while but found nothing .
1
vote
1answer
33 views
Greatest term of this expansion?
If in the expansion of $(x+a)^{15}$ the $11^{\text{th}}$ term is G. M. i.e. geometric mean of $8^{\text{th}}$ and $12^{\text{th}}$ terms then which term is the greatest term among the terms in ...
4
votes
2answers
73 views
Find the number of 5-member committees which include at least two Republicans
There are 10 Republicans, 8 Democrats and 2 Independent legislators eligible for committee membership.
How many 5-member committees exist which include at least two Republicans?
My Work :
$C(20,5)...
1
vote
2answers
29 views
If there are m items of one kind,n items of another kind and so on, then the number of ways of choosing r items of these items is
Please prove this result:
If there are m items of one kind,n items of another kind and so on, then the number of ways of choosing r items of these items is
=Coefficient of $x^r$ in $(1+x+x^2+...+x^...
3
votes
0answers
13 views
The number of ways in which n identical items can be divided into r groups so that no group contain less than m items and more than k$(m<k)$ is
Please prove this result
The number of ways in which n identical items can be divided into r groups so that no group contain less than m items and more than k$(m<k)$ is
=Coefficient of $x^n$ in ...
1
vote
4answers
104 views
I don't see how the binomial theorem relates to the principle of inclusion and exclusion?
I'm learning discrete maths as a hobby at the moment and I got stuck when the tutor starting relating the binomial theorem to the principles of inclusion and exclusion. The video I was watching is ...
0
votes
2answers
18 views
Find the coefficient of a negative indexed $x$ in a series expansion
Find the coefficient of x-10 in the expansion:
(2-1/x2)8
ANS: -448
I've tried using the General Term formula and got stuck at -x2r = x-10. Also, I tried expanding the equation but it doesn't look ...
-1
votes
2answers
54 views
Expansion of $(\frac{x+2}{x})^{-\frac{1}{2}}$ and then find an approximate value…
First, obtain the four terms in the expansion of $
(\frac{x+2}{x})^{-\frac{1}{2}}$ then let x = 100 and use the result
to find an approximate value of $(\frac{450}{51})^{\frac{1}{2}}$.
I am stuck ...
0
votes
2answers
38 views
$\sum_{k=0}^n \binom{n}{k}x^k(1-x)^{n-k}k$? [duplicate]
I cant wrap my head around simplifying the following sum:
$$\sum_{k=0}^n \binom{n}{k}x^k(1-x)^{n-k}k,$$
where $0<x<1$.
I tried to apply standard formulas here.
1
vote
3answers
59 views
Induction step in proof that $\binom{s}{s} + \binom{s + 1}{s} + \cdots + \binom{n}{s} = \binom{n + 1}{s + 1}$
Prove by induction that (binomial theorem)
$$
\binom{s}{s} + \binom{s+1}{s} + \dotsb + \binom{n}{s} = \binom{n+1}{s+1}
$$
for all $s$ and all $n>s$.
I used base case $s=0$, and I got my base case ...
4
votes
3answers
61 views
Prove that $3^{16} -33$ and $3^{15} +5$ is divisible by 4 by means of binomial theorem
This is a question that I found in a textbook:
Given that $p=q+1$, $p$ and $q$ are integers, then show that
$p^{2n} - 2nq-1$ is divisible by $q^2$ given that $n$ is a positive integer.
By taking a ...
2
votes
2answers
57 views
Express $\frac x{x^2-3x + 2}$ in the partial fraction form and show that its partial fraction $x^3$ can be neglected
I am having trouble solving a multi part question.
Express $ \frac x{x^2-3x + 2} $ in the partial fraction form.
The answer I got was $\frac2{x-2}-\frac1{x-1}$ .
The problem comes when they asked:
...
2
votes
1answer
58 views
Expand $\sqrt{1-x}$ up to and including the term $x^2$ (How should i proceed further)
I am trying to solve a 2 part question, the first part is to expand $\sqrt{1-x}$ up to and including the term $x^2$ which I did.
This gives me $1-(0.5)x-(0.125)x^2 + ...$
However, the 2nd part of ...
0
votes
1answer
33 views
When proving $\binom{n}{r} = \binom{n-1}{r} + \binom{n-1}{r-1}$ per induction over $n$, why don’t we need induction over $r$?
Proving this formula by induction is done by induction over $n$:
$$
\binom{n}{r} = \binom{n-1}{r} + \binom{n-1}{r-1}
$$
My question is: Why don't we need to show this seperately for $r \to r+1$ too?
2
votes
2answers
55 views
Binomial expansion lower bound $A^n + B^n \le (A+B)^n$ for non-integer $n$
By the Binomial expansion for integer powers,
$$ (A+B)^n = \sum_{k=0}^n {n\choose{k}} A^{n-k} B^{k}$$
(I'm assuming $A,B\ge 0$) and so we get the easy estimate $A^n + B^n \le (A+B)^n$ for any ...
4
votes
2answers
78 views
Prove $\sum_{r=0}^n \binom{n}{r} \binom{n+r}{r} (-2)^r =(-1)^n\sum_{r=0}^n \binom{n}{r}^2 2^r$ [closed]
Here is a combinatorics question that I am struggling.
Prove $\sum_{r=0}^n \binom{n}{r} \binom{n+r}{r} (-2)^r =(-1)^n\sum_{r=0}^n \binom{n}{r}^2 2^r$
I tried to simply the binomial coefficients on ...
1
vote
1answer
72 views
simplifying triple summations from series expansion
I have the following equation I would like to extract $x^k$ out of
$$
\sum^\infty_{k=0}x^k \Gamma \bigg(l,\frac{ax-1}{a^2}\bigg)
$$
I start by expanding the incomplete gamma function, and then the ...
4
votes
1answer
107 views
How to prove the two identities
Let $m,n$ be positive integers and $N=\{1,\ldots, n\}$. Try to prove the two following identites:
For $m<n$, we have
$$\sum_{A \subseteq N} (-1)^{\left| A\right|} \left(\sum_{j \in A} x_j\right)^m ...
0
votes
1answer
39 views
Induction based on combinations and binomial theorem: ${}^nC_0+{}^{n+1}C_1+{}^{n+2}C_2+\dots+{}^{n+p}C_p={}^{n+p+1}C_p$ [duplicate]
I was looking at some questions in a Cambridge text and I reached this question however I am at it for 1 hr and can't seem to get the proof right. Any help ?
1
vote
0answers
44 views
Could someone help clarify the meaning of this pre-calculus question on Binomial Expansion?
I'm taking an online pre-calculus course and I'm currently working on a unit about Binomial expansion.
I've come across an oddly worded (at least to me) question on coefficients:
"What coefficient ...
1
vote
1answer
63 views
A lower bound for $(1-e^x)^n$
I want to find a lower bound for
$$(1-e^x)^n$$
$n$ integer, $x$ real, and $1-e^x\geq 0$. One lower bound is (Bernoulli's inequality)
$$(1-e^x)^n\geq 1-ne^x$$ But I need a tighter lower bound that is ...
0
votes
2answers
42 views
What kind of expansion is this?
In a paper I came across an expansion like this:
$$\cos(m\theta) = C_m^0\cos^m(\theta) - C_m^2\cos^{m-2}(\theta)(1-\cos^2(\theta)) + C_m^4\cos^{m-4}(\theta)(1-\cos^2(\theta))^2 + ...
(-1)^nC_m^{2n}\...
0
votes
0answers
25 views
Generalizing $(^4C_0)^2-(^4C_1)^2+(^4C_2)^2-(^4C_3)^2+(^4C_4)^2=\,^4C_2$
I got this question, and I have a bit of an issue with part (b). I managed to prove the attached result; however, it does not really fit the above result (as it said generalise the result and then ...
0
votes
1answer
42 views
Sum of weighted binomial coefficients [duplicate]
I am struggling with computing the following sums:
$$\sum_{k=1}^{n}k\binom{n}{k}=\binom{n}{1}+2\binom{n}{2}+...+n\binom{n}{n}$$
and
$$\sum_{k=0}^{n}\frac{1}{k+1}\binom{n}{k}=\binom{n}{0}+\frac{...
2
votes
3answers
64 views
Writing $(m+2)^n-(m-2)^n$ in summation notation.
I have expanded $(m+2)^n-(m-2)^n$ the following way:
$$(m+2)^n-(m-2)^n = 2 {n \choose 1}m^{n-1}+ \dots + {n \choose n-1}m2^{n-1}-{n \choose n-1}m(-2)^{n-1}+{n \choose n}2^n - {n \choose n}(-2)^n$$
...
0
votes
1answer
86 views
A Subtle Point in Rudin's Proof on $e$
When Rudin proves the theorem $\lim_{n\rightarrow \infty} (1+\frac{1}{n})^n=e$, he uses the following claim:
Let $t_n= (1+\frac{1}{n})^n=1+1+\frac{1}{2!}(1-\frac{1}{n})+\frac{1}{3!}(1-\frac{1}{n})(...
0
votes
1answer
51 views
Polynomial of $n-1$ degree
Given a polynomial in $\mathbb{C}$, I want to show that a polynomial of the form
$$P_n(z)=\sum^n_{i=0} a_iz^i$$
can be decomposed into something like
$$P_n(z)=(z-z_0)\cdot Q_{n-1}(z),$$
where $Q$ is ...
2
votes
1answer
51 views
Which type of random walk has distribution of “scaled” binomial coefficient?
We know that, for a 1D symmetrical random walk, ${\displaystyle p = 1/2, q = 1/2}$, with equal walk step length, after n steps, its probability distribution will be proportional to binomial ...
0
votes
2answers
41 views
How to prove $n^p>\log n$ for a fixed $p\in (0,1)$ when $n\rightarrow \infty$? [duplicate]
I want to prove:
$$n^p>\log n$$
for a fixed $p\in (0,1)$ when $n\rightarrow \infty$.
I test by some big number and it seems true. But how to prove that?
