All Questions
299
questions
0
votes
1
answer
34
views
Binomial identity involving square binomial coefficient [closed]
I want to prove this identity, but I have no idea...
Could someone please post a solution? Thank you.
$$\sum_{k=0}^{n} \binom{-1/2}{n+k}\binom{n+k}{k}\binom{n}{k}= \binom{-1/2}{n}^2$$
(Maybe -1/2 can ...
3
votes
1
answer
45
views
Steps on solving part b of the bit-string exercise?
This is the exercise:
How many bit strings of length $77$ are there such that
a.) the bit
string has at least forty-six $0$s and at least twenty-nine $1$s, and also
the bit string corresponding to ...
- 131
2
votes
2
answers
47
views
Given a set of integers, and the number of summations find the resulting frequencies
Given a set $X = \{x_1,x_2,...x_m\}\subset \mathbb{Z}$ and the number of allowed addends $N$. How can I find the frequency of every possible sum?
Example: $X = \{-1, 2\}$ and $N=3$ then every ...
- 389
0
votes
0
answers
44
views
Choosing an ordered triplet of non-negative integers $(m_1, m_2, m_3)$ such that $m_1 + m_2 + m_3 = n$.
Define
$$A = \{ (m_1, m_2, m_3) : m_1 \geq 0, m_2 \geq 0, m_3 \geq 0, m_1 + m_2 + m_3 = n\}$$
Given that $n \geq 0$ and $n, m_1, m_2, m_3 \in \mathbb{Z}^+$, then why it is the case that
$$\vert A \...
- 934
0
votes
1
answer
27
views
Prove that $C(n,j)$ is periodic on $\mathbb{F}_2$? [duplicate]
Fix j and consider the sequence of binomial coefficients $[C(n,j)]_n$ on $\mathbb{F}_2$. It seems like it is periodic, but I am not sure how to prove it, or to determine its period.
- 23
6
votes
1
answer
147
views
How to find all positive integers $n,k$ such that ${n\choose k}=m$ for a given $m$?
This question is motivated by a simple exercise in Peter Cameron's Combinatorics: Topics, Techniques, Algorithms:
A restaurant near Vancouver offered Dutch pancakes with ‘a thousand and
one ...
- 2,454
0
votes
1
answer
66
views
Seeking help to Prove The Recursive Binomial Coefficient Formula
A rich body of research exists on Binomial Coefficients, with the concept finding its roots in Pascal's Triangle. My current investigation focuses on a Recursive Approach for generating these ...
0
votes
0
answers
22
views
How many bit strings of length n contain more 0’s than 1’s? [duplicate]
To solve this, I think we need to use combinatorial reasoning.=
Consider a bit string of length ( n ). There are ( 2^n ) possible bit strings of this length because each bit can independently be ...
- 147
0
votes
0
answers
38
views
find number of disjoint subsets
For my discrete maths course i did the following exercise:
Let M be a finite set with n elements, find
$|\{(U,V)|U,V\subseteq M , U \not = V,U\cap V= \emptyset \}|$
I did it the following way:
choose ...
- 37
3
votes
1
answer
67
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 $ ...
2
votes
1
answer
75
views
In how many ways it is possible to take out balls from the basket , such that will take out at least one from each color?
The question:
In a basket there are $20$ black balls, $15$ white balls and $18$ red balls. All balls with the same color are identical to each other. In how many ways it is possible to take out balls ...
5
votes
1
answer
214
views
Approximating a sum of products of binomial coefficients
I have a function I want to approximate the growth of in a closed form:
$$\frac{\sum_{i=0}^{n-1}\sum_{j=0}^{n} |n-i-j|\binom{n+i-j}{i}\binom{n-i+j-1}{j}}{\binom{2n}{n}}$$
Without the absolute value ...
- 69
0
votes
1
answer
68
views
How many lattice paths from $(0, 1, 2)$ to $(5, 5, 5)$ that pass through $(3, 3, 3)$ but not $(1, 2, 3)$?
How many lattice paths from $(0, 1, 2)$ to $(5, 5, 5)$ that pass through $(3, 3, 3)$ but not $(1, 2, 3)?$
My answer
Firstly try to get how many possible shortest lattice paths from $(0, 1, 2)$ to $(3,...
- 11
0
votes
0
answers
24
views
Sum of $\binom ni$, where $i$ goes from $0$ to $k < n$ [duplicate]
I wonder what the value of this sum is:
$$S = \sum_{i=0}^k \binom ni$$
When $k=n$, then $S = 2^n$, but what if $k<n$?
- 1,129
1
vote
2
answers
85
views
Prove that $\sum\limits_{i=0}^n(-1)^i\binom{n}{i}\binom{kn-ki}{n-1}=0$
Given $$\sum\limits_{i=0}^n(-1)^i \binom{n}{i}\binom{m-ik+n-1}{n-1}$$ (which can be interpreted as the number of solution sets to the equation $x_1+x_2+\cdots+x_n=m$ where $0≤x_i<k$), prove that $$\...
- 637