# Questions tagged [binomial-coefficients]

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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### How do you find the coefficient of $x$ in $(x + 1)^2$?

I want to learn how can I find out the coefficient of the variable $x$ in the expression $(x + 1)^2$. It is a case of a perfect square expansion.
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### Sum of the product of binomial coefficients

I am trying to prove $\sum_{j=1}^{n-k}{n \choose j}{n-k-1 \choose j-1}={2n-k-1 \choose n-k}$. I tried to apply Vandermonde's Identity, however I have not been able to.
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### How was this identity found: $\sum_{k=1}^{n}\binom{n}{k}\sum_{1\le i\le j\le k}\frac{1}{ij}=\sum_{1\le i\le j\le n}\frac{2^n-2^{n-i}}{ij}$?

For any postive integer $n$, prove $$\sum_{k=1}^{n}\binom{n}{k}\sum_{1\le i\le j\le k}\dfrac{1}{ij}=\sum_{1\le i\le j\le n}\dfrac{2^n-2^{n-i}}{ij}.$$ This problem is 2019 AMM problem. I don't ask ...
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### Show $\sum_{k=1}^n {k+1\choose 2}{2n+1\choose n+k+1}={n\choose 1}4^{n-1}$

I've been attempting to show that: $$\sum_{k=1}^n {k+1\choose 2}{2n+1\choose n+k+1}={n\choose 1}4^{n-1}\\ \sum_{k=2}^n {k+2\choose 4}{2n+1\choose n+k+1}={n\choose 2}4^{n-2}$$ Can anyone give some ...
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### How to interchange the two summations and simplify?

How can we simplify the following summation? $$\sum_{i=0}^{r-j}\sum_{l=0}^{i+j-1}(-1)^{i-1}\binom{i+j}{i}\binom{r-i-1}{r-j-i}F(r,j,l).$$ Here, $F(r,j,l)$ is independent of $i$. So, how are the limits ...
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### Minimum n in order to have at least C combinations of order r?

I am a bit stuck on this problem. How do I find the minimum number of elements n which ensure that I have at least C combinations of order r ? Basically, for a given C and r, find the minimum n which ...
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### Proof of summation converging to 1 [duplicate]

Can someone prove this?$$\sum_{k=n}^{2n} 2^{-k}\dbinom{k}{n}=1$$
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### Represent $\sum_{k=1}^{n} k^{2}$ in terms of binomial coefficient.

Came across a probability problem that is sort of challenging for a beginner in a sense that I may have not seen or came across a lot of binomial identities. What I am looking for is to see if there ...
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### combination problem with addition [closed]

Please tell me what are the ways getting 37 by adding 6 digit using 1 to 17.No restrictions.
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### How would I represent combinations of fixed subsets?

Given a friendship group of $n$ people, the binomial coefficient ${n \choose k}$ can tell me how many combinations of $k$ friends there are who could meet, e.g., if there are $10$ people and we meet ...
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### Solving $1+2+3+4+ \cdots+ n=$n(n+1)/2)through use of binomial coefficient

I am trying to understand proofs of the $1+2+3+4+ \cdots$ series. I'm puzzled by the 3rd point of this post where it is solved by binomial coefficient https://math.stackexchange.com/a/2288/777575 ...
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### Calculating the number of poker hands that have exactly 4 different denominations

I'm been working through some examples of questions mentioned in the title and came across this one that I wasn't entirely if I was approaching this correctly. My initial approach to solving this ...
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### Combinatorial identity: $\sum\limits_{k=0}^{i\land j}\binom ik(-1)^k\binom{i+j-k}i=1$

Let $i,j\in\mathbb Z_{\ge0}$ be nonnegative integers. How can we prove $$\sum_{k=0}^{i\land j}\binom ik(-1)^k\binom{i+j-k}i=1?$$ (Here, $i\land j=\min(i,j)=\min\{i,j\}=\min(\{i,j\})$ is the minimum of ...
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### Show that $\binom{n-1}{r-1}=\sum^{r-1}_{k=0}(-1)^k\binom rk\binom{n+r-k-1}{r-k-1}$
This is an identity that I couldn't prove. $$\binom{n-1}{r-1}=\sum^{r-1}_{k=0}(-1)^k\binom rk\binom{n+r-k-1}{r-k-1}.$$ I believe that the identity is provable through combinatorics, but I can't ...
### Prove or disprove $\frac{(x+n)!}{(x!)\text{lcm}(x+1, \dots, x+n)} < (n-1)!$
Is it correct that for any positive integers $x,n$, that $\frac{(x+n)!}{(x!)\text{lcm}(x+1, \dots, x+n)} < (n-1)!$ where lcm is the least common multiple. I ask because I find this relationship ...