Coefficients involved in the Binomial Theorem. $\binom{n}{k}$ counts the subsets of size $k$ of a set of size $n$.
1
vote
2answers
43 views
Why is the binomial coefficient $C(-1,k)=(-1)^k$?
It is known that $$C(n,k) = \frac{n^{k<}}{k!},$$ where $C(n,k)$ is the notation for the binomial coefficient and $k<$ is notation for '$k$ lower' which gives
$$n(n-1)(n-2)....(n-k+1).$$ For $n = ...
0
votes
1answer
54 views
Calculating a recursive power term binomial sum
Could someone please help me or give me a hint on how to calculate this sum:
$$\sum_{k=0}^n \binom{n}{k}(-1)^{n-k}(x-2(k+1))^n.$$
I have been trying for a few hours now and I start thinking it may ...
12
votes
4answers
292 views
How to prove $\sum\limits_{k=0}^n{n \choose k}(k-1)^k(n-k+1)^{n-k-1}= n^n$?
How do I prove the following identity directly?
$$\sum_{k=0}^n{n \choose k}(k-1)^k(n-k+1)^{n-k-1}= n^n$$
I thought about using the binomial theorem for $(x+a)^n$, but got stuck, because I realized ...
-1
votes
1answer
29 views
Formula issues when working out chances of getting certain marks [duplicate]
$$P(X = k) = \binom{N}{k} (0.5)^k (0.5)^{N-k} = \binom{N}{k} (0.5)^N$$
Using formula above, I have got the following results for chances for getting certain percentage on a $50$ question paper, each ...
0
votes
3answers
66 views
How to correctly write a binomial distribution for a $50$ questions exam [duplicate]
Using binomial distribution I want to know what is the chance of getting $70\%$ or greater in a $50$ question exam, each question having a true/false option to select from. What is the correct formula ...
1
vote
1answer
91 views
Sum of the first k binomial coefficients for fixed n
I am reading Remarks on a Ramsey theory for trees by Janos Pach, Jozsef Solymosi and Gabor Tardos.
Let $k, d, n \geq 2$ be integers. Somethig interesting happens when $$2^{n/k} > \sum_{i=0}^{d-1} ...
1
vote
2answers
81 views
Probability of getting >70% in exam with 50 yes/no questions
In a paper containing 50 yes/no questions, I am trying to find the probability of getting 70%.
Using binomial distribution,
$$P(X\ge70\%)=\sum_{k=25}^{50} \binom{50}{k}\left(\frac{1}{2}\right)^{50}$$
...
13
votes
3answers
411 views
Proving that $\sum_{i=0}^{n}\binom{n}{i}i^{n-i}(n-i)^{i}\le\frac{1}{2}n^n$
How can we prove that
$$\displaystyle\sum_{i=0}^{n}\binom{n}{i}i^{n-i}(n-i)^{i}\le\dfrac{1}{2}n^n$$
where $\displaystyle\binom{n}{i}=\dfrac{n!}{i!(n-i)!}$.
This inequality is very interesting. I ...
0
votes
1answer
85 views
Probability generating function of the negative binomial distribution.
I am using the definition of the negative binomial distribution from here. This is the same definition that Matlab uses. For convenience,
$$P(k) = {r + k -1 \choose k}p^r(1-p)^k ,$$
where $p$ is ...
36
votes
5answers
1k views
Solutions to $\binom{n}{5} = 2 \binom{m}{5}$
In Finite Mathematics by Lial et al. (10th ed.), problem 8.3.34 says:
On National Public Radio, the Weekend Edition program posed the
following probability problem: Given a certain number of ...
0
votes
0answers
31 views
is this binomial distribution correct?
I am trying to work out what is the chances of getting the following marks $(100\%, 70\%, 60\%, 50\%)$ in a paper containing $50$ questions, each question containing yes/no options.
Using Binomial ...
0
votes
1answer
30 views
Are binomial distribution answers expressed as percentages?
I have a question on an answer to a binomial distribution question. The chances of getting $100\%$ in a test is $8.881784197 \cdot 10^{-16}$. Is that actually a percentage? Making it ...
0
votes
3answers
58 views
Binomial coefficients (1/2, k)
I don't understand questions that involve a binomial expression where you have a fraction choose k or a negative number choose k. I understand and am able to do it when there are no fractions and they ...
0
votes
1answer
36 views
Sample size for Wilsons confidence interval
Consider a story ranking website in which the ranking is crowd sourced from the number of up votes and down votes received by a story.
The score is computed as the lower bound Wilson's algorithm.
...
0
votes
0answers
32 views
Value of $\quad\frac{\sum_{r=0}^{24}\binom{100}{r}.\binom{100}{4r+2}}{\sum_{r=1}^{25}\binom{200}{8r-6}}= $
Value of $\displaystyle \frac{\sum_{r=0}^{24}\binom{100}{r}.\binom{100}{4r+2}}{\sum_{r=1}^{25}\binom{200}{8r-6}}= $
My try:: $\displaystyle \sum_{r=0}^{24}\binom{100}{r}.\binom{100}{4r+2} = ...
1
vote
2answers
66 views
Sum of the series $\sum_{k=0}^{r}(-1)^k.(k+1).(k+2).\binom{n}{r-k} $
for $n>3$, The sum of the series $\displaystyle \sum_{k=0}^{r}(-1)^k.(k+1).(k+2).\binom{n}{r-k} = $
where $\displaystyle \binom{n}{r} = \frac{n!}{r!.(n-r)!}$
My try:: I have expand the ...
1
vote
2answers
164 views
Prove that $\sum\limits_{i=1}^n 2i\binom{2n}{n-i}= n\binom{2n}{n}$
Let n be a positive integer.
Prove that $$\sum_{i=1}^n 2i\binom{2n}{n-i}= n\binom{2n}{n}$$
16
votes
1answer
303 views
A Combinatorial Proof of Dixon's Identity
Dixon's Identity states:
$$ \sum_{k} (-1)^k\binom {a+b}{b+k}\binom{b+c}{c+k}\binom{c+a}{a+k} = \binom{a+b+c}
{a,b,c}$$
A bit of history:
The case $a=b=c$ was proved by Dixon in 1891 using ...
1
vote
2answers
40 views
Summations with binomial coefficients:$\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}$
Can someone help me solve this equation? How to prove that $$\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}?$$
0
votes
0answers
18 views
Iterated Vandermonde identity
Let $r,s \in \mathbb{C}$ or more generally elements of a commutative $\mathbb{Q}$-algebra $R$ and $n \in \mathbb{N}$. Then we have the following "iterated Vandermonde identity":
$$\binom{r\cdot s}{n} ...
1
vote
3answers
64 views
Simplify the Expression $\sum _{ k=0 }^{ n }{ \binom{n}{k}}i^{k}3^{k-n} $
I should simplify the following expression (for a complex number):
$$\sum _{ k=0 }^{ n }{ \binom{n}{k}}i^{k}3^{k-n} $$
The solution is $(i+\frac{1}{3})^n$,but i don't quite get the steps. If would be ...
2
votes
0answers
33 views
Characterization of $\lambda,\mu\vdash n$ for which $\displaystyle{n\choose\mu}\mid{n\choose\lambda}$
In view of this question, I was wondering about general characterizations of $\mu,\lambda\vdash n$ for which
$${n\choose\mu}\,\left|\,{n\choose\lambda}\right.,$$
(see multinomial coefficient and ...
0
votes
1answer
14 views
Number of partitions of positive integer $k$ into $f$ non zero integers
In statistical physics my teacher said me the no of partitions of a positive integer $k$ into $f$ partitions is same as no of ways arranging $f$ gaps on a total of $(k+f)$ sites (equals to ...
6
votes
1answer
85 views
Vandermonde identity in a ring
Let $R$ be a commutative $\mathbb{Q}$-algebra. For $r \in R$ and $n \in \mathbb{N}$ we can define the binomial coefficient $\binom{r}{n}$ as usual by $\binom{r}{0}=1$ and ...
6
votes
3answers
125 views
How many solutions are possible to this equation?
Given
$$A+2B+3C=N
$$
where $N$ is a given positive integer.
$A ,B,C\in\mathbb{N}$ vary from $0$ to $\infty$.
How many solutions will be there to this equation?
2
votes
1answer
52 views
Bernstein Polynomials and Expected Value
The first equation in this paper
http://www.emis.de/journals/BAMV/conten/vol10/jopalagzyl.pdf
is:
$$\displaystyle B_nf(x)=\sum_{i=0}^{n}\binom{n}{i}x^i(1-x)^{n-i}f\left(\frac{i}{n}\right)=\mathbb E ...
6
votes
0answers
75 views
Binomial formula for $(x+1)^{1/3}$ (related to Newton's binomial theorem)
I know that
$$\displaystyle \sqrt{1+x} = \sum_{j=0}^{\infty}\left( \frac{(-1)^{(j-1)}}{2^{2j-1}\cdot(2j-1)}\binom{2j-1}{j}x^j\right). $$
Now, I want to evaluate $\sqrt[3]{1+x}$ but stuck at some ...
2
votes
3answers
73 views
Combinatoric Explanation of General Identity
When $k \lt n$, what is the value of the sum $$\sum\limits_{j=0}^n {n \choose j}(-1)^j (n-j)^k.$$ Explain combinatorially.
Any ideas on where to start?
0
votes
1answer
37 views
What is the probability of drawing kings
A hand H of 5 cards is chosen randomly from a standard deck of 52. Let $E_1$ be the
event that H has at least one King and let $E_2$ be the event that H has at least 2 Kings.
What is the conditional ...
0
votes
2answers
54 views
Sum with binomial coefficients and fraction
Is there a closed form known for
$$
\phi(n,a,c)=\sum_{k=0}^n \binom{n}{k} a^k \frac{1}{k+c}
$$
where $a <0$, $c>0$ and $n \in \mathbb{N}$?
I know the answer for two special cases:
$$
...
11
votes
3answers
249 views
Closed form for $\sum_{k=0}^{n} k\binom{n}{k}\log\binom{n}{k}$
Is it possible to write this in closed form:
$$\sum_{k=0}^{n} k\binom{n}{k}\log\binom{n}{k}$$
Can you get something like $$n2^{n-1}\log(2^{n-1})$$
1
vote
1answer
15 views
how to get a binomial from a summation
An urn contains 6 Red balls and 1 Blue ball. A fair die having faces f1;2;3;4;5;6g is
rolled. If the top face on the die shows m, then m random balls are removed from the urn.
What is the expected ...
0
votes
2answers
33 views
Relation between binomials
how can I prove that the following relation is true:
$$\binom{x-2}y+2\binom{x-2}{y-1}+\binom{x-2}{y-2}=\binom{x}y$$
Thank you for hints or references!
Marted
9
votes
1answer
107 views
Can this product be written so that symmetry is manifest?
Let $i,$ $j,$ $k$ be nonnegative integers such that $i+j+k$ is even. The expression
$$(-1)^{j+k}\binom{i+j+k}{i,j,k}\prod_{\ell=0}^{k-1} \frac{i-j+k-2\ell-1}{i+j+k-2\ell-1}$$
apparently computes the ...
5
votes
1answer
78 views
How to compute the sum of every $k$-th binomial coefficient?
My teacher was discussing binomial expansions of $(1 + x)^n$ and he gave as an interesting example with $x = i$ whereby you could obtain the sum of all the odd coefficients ($C_n^1+ C_n^3+ C_n^5 ...$) ...
2
votes
1answer
57 views
Theta asymptotic for $\binom{2m}{m}$ [duplicate]
Show that
$\binom{2m}{m} = \Theta\left(\frac{2^{2m}}{\sqrt{m}}\right)$ without using Stirling's approximation.
2
votes
2answers
58 views
Calculating an “at least” probability without summation?
I know One can calculate the probability of getting at least $k$ successes in $n$ tries by summation: $$\sum_{i=k}^{n} {n \choose i}p^i(1-p)^{n-i}$$
However, is there a known way to calculate such ...
10
votes
7answers
303 views
Summation simplification $\sum_{k=0}^{n} \binom{2n}{k}^2$
$\sum_{k=0}^{n} \binom{2n}{k}^2$
So i'm trying to simplify this one and I'm stuck in nowhere. Some kind of tip would be appreciated.
Thanks! :)
2
votes
1answer
63 views
Why the nth power of a Jordan matrix involves the binomial coefficient?
I've searched a lot for a simple explanation of this. Given a Jordan block $J_k(\lambda)$, its $n$th power is:
$$
J_k(\lambda)^n = \begin{bmatrix}
\lambda^n & \binom{n}{1}\lambda^{n-1} & ...
5
votes
1answer
80 views
$\sum_{i=0}^m \binom{m-i}{j}\binom{n+i}{k} =\binom{m + n + 1}{j+k+1}$ Combinatorial proof
Is there a simple combinatorial proof for the following identity?
$$\sum_{0\leq i \leq m} \binom{m-i}{j}\binom{n+i}{k} =\binom{m + n + 1}{j+k+1}$$
where $m,j \geq 0$, $k \geq n \geq 0$.
0
votes
2answers
84 views
How to show $\binom{2p}{p} \equiv 2\pmod p$?
how to prove $\forall p$ prime :
$\binom{2p}{p} \equiv 2 \pmod p$ we have:
$\binom{2p}{p} = \frac{2p (2p-1)(2p-3)...1}{p!p!}$ but how to continue?
0
votes
1answer
81 views
Using the general binomial theorem to find a series-like expression for $\sqrt 2$
How do I use the general binomial theorem (i.e. the series expansion of ${(1+x)^\alpha}$ for $ |x|<1$) to show the following? $$\sqrt 2=1+\frac 1{2^2}+\frac{1\cdot3}{2!\cdot{2^4}} ...
2
votes
2answers
76 views
Stirling's Approximation
A sharp Stirling's approximation form states that $$n! \sim \left(\frac{n}{e}\right)^n\sqrt{2\pi n}.$$
Use that form to show that:
$$\binom{2m}{m} = \Theta\left(\frac{2^{2m}}{\sqrt{m}}\right).$$
1
vote
1answer
54 views
Prove the following identity combinatorially
Prove the following identity combinatorially
$$\left(\begin{array}{c} \left(\begin{array}{c} n \\ 2 \end{array}\right) \\ 2 \end{array}\right) = 3 \left(\begin{array}{c} n \\ 4 \end{array}\right) + ...
0
votes
2answers
63 views
Why is $C(n + r − 1, r) = C(n + r − 1, n − 1)$, specifically why is $r$ equivalent to $n-1$?
I have this theorem in my discrete math textbook:
There are $C(n + r − 1, r) = C(n + r − 1, n − 1)$ r-combinations from a set with n elements when repetition of elements is allowed.
I can't figure ...
4
votes
2answers
101 views
Prove by Combinatorial Argument that $\binom{n}{k}= \frac{n}{k} \binom{n-1}{k-1}$
This is a question from my first proofs homework and I am confused about the combinatorial argument aspect. I already did the algebraic proof. I think I am supposed to put into words what both sides ...
1
vote
1answer
96 views
Arrangements with No Two Vowels Consecutive
In general we state that there are ${r-wn - (n-1) \choose (n-1)}$ ways to distribute r identical balls in n distinct boxes with at least w balls in each box. Considering this, how many ways are there ...
4
votes
2answers
255 views
Inductive proof that ${2n\choose n}=\sum{n\choose i}^2.$
I would like to prove inductively that $${2n\choose n}=\sum_{i=0}^n{n\choose i}^2.$$
I know a couple of non-inductive proofs, but I can't do it this way. The inductive step eludes me. I tried naively ...
3
votes
2answers
74 views
How to get the sum of the values in a $N \times N$ table?
How to get the sum of the values in a $N \times N$ table (without adding repeating products such as $6 \times 7$ and $7 \times 6$ twice and without counting perfect squares)?
Figured out that
$1 ...
2
votes
1answer
59 views
Choosing k Multisets from [n]
We are to play a lottery game where five numbers are drawn out of [90], but the numbers drawn are put back into the basket right after being selected. To win the jackpot, one must have played the same ...
