Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [binomial-coefficients]

For questions involving the coefficients involved in the binomial theorem. $ \binom{n}{k}$ counts the subsets of size $k$ of a set of size $n$.

The binomial coefficient $\binom{n}{k}$ can be defined in several equivalent ways for $n$ and $k$ non-negative integers:

  1. The number of subsets of size $k$ of a set of size $n$.
  2. Element $k$ of row $n$ in Pascal's triangle (counting the first element or row as $0$).
  3. $\dfrac{n!}{k!(n-k)!}$
  4. The coefficient of $x^k$ in $(1+x)^n$.

The binomial theorem says that $$(x+y)^n=\sum_{k=0}^n\binom{n}{k}x^{n-k}y^k$$ using the convention that $0^0=1$.

Binomial coefficients can be extended for arbitrary complex $\alpha$ through the formula: $$\binom{\alpha}{k}=\frac{\alpha(\alpha-1)(\alpha-2)\dots(\alpha-k+1)}{k(k-1)(k-2)\dots1}$$

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Upper bound of binomial distribution

I'm studying a proof and I'm wondering which binomial approximation could have been use to establish the following bound: $$\cfrac{1}{2} {n \choose r} {n-r \choose r} \le \cfrac{n^{2r}}{2(r!)^2}$$ I get that: $$\cfrac{1}{2} {n \choose r} {n-r…
MasterScrat
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Show that $\binom{n}{1}-\cdots+(-1)^{n-1}\binom{n}{n}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\right)=\frac{1}{n}$

Show that $$ \sum_{r=1}^{n} (-1)^{r-1} \binom{n}{r} \left(1 + \frac{1}{2} + \cdots + \frac{1}{r} \right) = \frac{1}{n} $$ My attempt. I tried by taking the general term $$ T_r = (-1)^{r-1} \binom{n}{r} \left(1 + \frac{1}{2} + \cdots + \frac{1}{r}…
hinsberg
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Proof that $\sum_{k=0}^n \frac{(-1)^k}{k+1} {n \choose k} = \frac{1}{n+1}$ for every $n \in N$.

I want to show that $$ \sum_{k=0}^n \frac{(-1)^k}{k+1} {n \choose k} = \frac{1}{n+1} $$ I know, that I will have to use that for $ l \geq 1 $ $$ \sum_{i=0}^l (-1)^i {l \choose i} = 0 $$. My idea is that I write: $$ \sum_{k=0}^n \frac{(-1)^k}{k+1} {n…
Akimbo_Slice
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For integers $n,r$, let $\binom nr = \begin {cases} \binom nr & n\ge r\ge 0 \\ 0 & \text{otherwise} \end {cases}$

For integers $n,r$, let $\binom nr = \begin {cases} \binom nr & n\ge r\ge 0 \\ 0 & \text{otherwise} \end {cases}$. Find the maximum value of $k$ for which the sum $\sum_{i=0}^k \binom {10}{i} \binom{15}{k-i} +\sum_{i=0}^{k+1} \binom {12}{i} \binom…
Aditya
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How do you find the independent term in $(x + 1 + x^{-1})^4$?

I know you could expand it all out and take cases but is there a way to group it such that there is a general term? I've been trying to group it in order to establish a general term ie something like $((x + 1) + x^{-1})^4$ to make a binomial. Any…
user71207
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Formula for row index of a minimal distance matrix?

I am working on the following problem: given a matrix of M rows, I calculate a minimal distance matrix containing the absolute differences among all the possible couples of rows. In order to make the distance matrix as minimal as possible, I do…
1Z10
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Find maximum and minimum value of: $\binom{16}{k}+\binom{16}{k-1},k\in N$

Find maximum and minimum value of: $\binom{16}{k}+\binom{16}{k-1},k\in N$ Now, $\binom{16}{k}+\binom{16}{k-1}$ is same as $\binom{17}{k}$ I think that minimum value would be for $k=17$, which is $1$. But what would be the maximum value?
untitled
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Binomial coefficients with sums

I need help solving this task, if anyone had a similar problem it would help me. The task is: Calculate: $$\sum\limits_{k=0}^{n} { 3n \choose 3k } $$ I tried something, with $$2^n= \sum\limits_{k=0}^{n} { n \choose k }$$ But, have no idea how to…
LogicNotFound
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Pascal's triangle, estimate row value by fixed row and maximum yields.

let res be the max yields. Let col be a fixed positive integer. Let $f(row)=\binom{\text{row}}{\text{col}}=A$, what is the largest value for row such that $f(row) \leq A$? Standard Pascal's triangle expression: $\binom{\text{row}}{\text{col}} =…
Weilory
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Binomial coefficients-sums

I need help solving this task so if anyone had a similar problem it would help me. Calculate: $\sum\limits_{i=0}^{n}(-1)^i i {n \choose i}$ I tried this: $\sum\limits_{i=0}^{n}(-1)^i i \frac{n!}{i!(n-i)!}\\\sum\limits_{i=0}^{n}(-1)^i…
LogicNotFound
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Approximation of $\frac {n-c \choose k} {n\choose k} $ using a radical

Is there a good approximation for $\frac {n-c \choose k} {n\choose k} $, given large parameter n (k,c can be large as well, but k,c
user3563894
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$ \sqrt[n]{ \binom{n}{1} \cdot \binom{n}{2} \cdot \binom{n}{2} ........... \binom{n}{n} }$

As a part of a limit question I was evaluating, it was required that I evaluate this sum. However I do know no easy closed forms of this. I have tried applying the vandermonde identity but what is messing me up is the index of summation: i.e: $$…
tryst with freedom
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Show that ${n\choose k} = {n\choose l} \iff n=k+l$

I was given the following problem: Let $n,k,l \in \mathbb{N}$ such that $0\le l< k \le n $. Show that, $${n\choose k} = {n\choose l} \iff n=k+l$$ I'm having trouble with the implication from left to right. I tried to prove it by contradiction…
George25
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$\sum_{i=0}^n (-1)^k \cdot C(n,k)$

Use the Binomial Theorem to Show $$\sum_{i=0}^n (-1)^k \cdot C(n,k)$$ I'm not sure where to start here . . . I know it is missing an $a^{n-k}$ and a $b^{k}$ term that maybe I should set to be 1?
rbtLong
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A pascal's triangle problem

I am trying to solve this pascal's triangle problem. It asks to this: "Moving only up or to the right, how many paths exist from point A to point B?" I used the standard method to solve this and got a maze looking like this: I got the answer 10 and…
Andy Johnson
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