# Tagged Questions

Coefficients involved in the Binomial Theorem. $\binom{n}{k}$ counts the subsets of size $k$ of a set of size $n$.

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### Combinatorial interpretation of multinomial function. [closed]

Given $n$ items if we pick $k$ we use binomial function. What is the analogy with multinomial function?
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### Binomial theorem question. Find the value of the constant $k$

$$\left[(k+x)\left(2-\frac{x}{2}\right)\right]^6$$ where the coefficient of $x^{2}$ is $84$.Find the value of the constant $k$. I tried to expand the equation but got a equation of degree 6 for some ...
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### # of bit strings of length n (even>2), with n/2-1 zeros and n/2+1 ones, zero followed by one

case 1: What is the number of bit strings of length 4, with 1 zero and 3 ones, zero must be followed by one Answer: 3 case 2: What is the number of bit strings of length 6, with 2 zeros and 4 ones, ...
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### Checking an identity involving binomial coefficients

I need some help to check the following identity: for every $0\leq i\leq l\leq r$ $$\sum_{j=0}^i\binom{r-l+i-j}{i-j}\binom{l-i+j}{j}=\binom{r+1}{i}.$$ Is this true ? Answering to John, this ...
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### Show that $\sum_{r=0}^{n}\binom{n}{r}\binom{m+r}{n}= \sum_{r=0}^{n}\binom{n}{r}\binom{m}{r}2^r$

Here $\binom{a}{b}$ is the number of ways in which $b$ objects can be chosen from a collection of $a$ distinct objects. Show that: \binom{n}{0}\binom{m}{n}+\binom{n}{1}\binom{m+1}{n}+\binom{n}{2}\...
Theorem: For any pair of positive integers $n$ and $k$, the number of $k$-tuples of positive integers whose sum is $n$ is equal to the number of $(k − 1)$-element subsets of a set with $n − 1$ ...
Consider the field $\mathbb{F}_q$ where $q=p^k$ for some prime $p$. I have some identities related to binomial coefficients over such a field, which I wish to prove. So, can someone tell me a source ...