All Questions
Tagged with binomial-coefficients combinatorics
3,224
questions
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16
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Find the number of lattice paths weakly under a slope $y = \mu x$
How many lattice paths are there from an arbitrary point $(a,b)$ to another point $(c,d)$ that stay weakly (i.e. it can touch the line) under a slope of the form $y = \mu x$, with $\mu \in \mathbb{N}$?...
- 11
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32
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Probability involving identical objects. I am not able to understand how the Ncr formula is being applied below for counting identical objects.
Question: A bag contains 5 identical red coins, 6 identical yellow coins and 8 identical blue coins. If 3 coins are picked up randomly from the bag, what is the probability that there is at least one ...
- 1,050
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2
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61
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binomial distribution but sometimes the last outcome doesn't matter
Here's the motivation for my question: I'm designing an RPG. To simplify as much as possible, lets say my enemy has $h = 4$ HP and I deal $a = 1$ damage with every attack.
However, there's also a $p$ ...
- 23
3
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3
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152
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Iterated rascal triangle row sums
In this manuscript the authors propose the following conjecture (1)
\begin{align*}
\sum_{k=0}^{4i+3} \binom{4i+3}{k}_i &= 2^{4i+2}
\end{align*}
where $\binom{4i+3}{k}_i$ is iterated rascal ...
- 558
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1
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61
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What is $\sum_{m=0}^{\lfloor k/3\rfloor}{2k\choose k-3m}$ [closed]
With the floor function, I am not sure how to approach this.
Edit: I have a formula $\frac{1}{6}\left(4^k+2+3\frac{(2k)!}{(k!)^2} \right)$, but I got it purely from guess work.
- 99
2
votes
2
answers
46
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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
1
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1
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37
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Summation of n-simplex numbers
Gauss proved that every positive integer is a sum of at most three triangular(2-simplex) numbers. I was thinking of an extension related to n-simplex.
Refer: https://upload.wikimedia.org/wikipedia/...
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45
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Closed form for nested sum involving ratios of binomial coefficients
I ran into the following nested sum of binomial coefficients in my research, but I couldn't find the closed form expression for it. I looked at various sources and still couldn't find the answer. So I ...
3
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1
answer
79
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Combinatorial Proof of the $\sum_{k=1}^{n} k^2 \binom{n}{k} = n(n + 1) 2^{n - 2}$ [duplicate]
How to prove that $$\sum_{k = 1}^{n}k^2\binom{n}{k} = n(n + 1)2^{n -2}$$ in an combinatorial way?
I have a algebraic proof method, but I don't know how to use a combinatorial proof method to do it.
- 31
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41
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Evaluating Difference of Product of Binomial Coefficients
As part of my project I'm asked to evaluate the positivity of the following difference: $$\binom{l_1+t-1}{l_1}\binom{l_2+m+t-1}{l_2+m}\sum_{j=0}^{t}\binom{t-j+m}{m}\binom{j+l_1}{j}\binom{j+l_2}{j}-\...
- 31
1
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1
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107
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Spivak Exercise, Prove Vandermonde's Identity $\sum_{k=0}^{l}\binom{n}{k}\binom{m}{l-k}=\binom{n+m}{l}$
Prove that
$$\sum_{k=0}^{l}\binom{n}{k}\binom{m}{l-k}=\binom{n+m}{l}$$
Hint: Apply the binomial theorem to $(1+x)^n(1+x)^m$
Proof:
Following Spivak's advice, we have
$\sum_{k=0}^{n}\binom{n}{k}x^k=(...
- 115
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44
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Choosing an ordered triplet of non-negative integers $(m_1, m_2, m_3)$ such that $m_1 + m_2 + m_3 = n$.
Define
$$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
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2
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44
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Trying to prove equivalence of combinatorial formula and nested summations
I’m sorry if this is a dumb problem, but I’m trying to get into mathematics and prove this but I’m only in 9th grade and haven’t found any sources on this:
Prove that $$ {n \choose r} = \overbrace{ \...
4
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3
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70
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$(1-x)^{n+a} \sum_{j=0}^\infty \binom{n+j-1}{j}\binom{n+j}{a} x^j = \sum_{j=0}^a \binom{n}{a-j}\binom{a-1}{j} x^j$
Let $n$ and $a$ be natural numbers. How to prove the following for $x \in [0, 1)$?
$$
(1-x)^{n+a} \sum_{j=0}^\infty \binom{n+j-1}{j}\binom{n+j}{a} x^j = \sum_{j=0}^a \binom{n}{a-j}\binom{a-1}{j} x^j
$$...
- 9,683
4
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1
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139
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What is the name of this combinatorial identity?
In the course of my physics research, I appear to have stumbled onto the following combinatorial identity:
$${dn\choose m}=\sum_{\vec k} {n \choose \vec k}\,\prod_{j=0}^d {d \choose j}^{k_j},$$
where $...
- 1,835