# Tagged Questions

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

33 views

43 views

### Binomial expansion of $(x_{1}+x_{2}+…+x_{k})^{n}$ [duplicate]

If we expand $$(x_{1}+x_{2}+...........+x_{k})^{n}$$ How many terms will be there once we collect terms with equal monomials? What is the sum of all coefficients? I literally have no clue how to ...
53 views

### how many terms will there be once you collect terms with equal monomials? What is the sum of all the coefficients?

If you expand $(x_1+x_2+\cdots+x_k)^n$, how many terms will there be once you collect terms with equal monomials? What is the sum of all the coefficients? I'm kind of lost here. This came up with ...
63 views

### Looking for a nonrecursive formula for the general derivatives of the quotient of functions

I want to prove that the $k$-th derivative $h^{(k)}(x)$ of the function $h(x)=\frac{1}{1+x^2}$ is zero at $x=0$ for all integer values $k>0$. My only idea was to go the stubborn way applying ...
35 views

### Find $\sum r\binom{n-r}{2}$

Let $A=\{1,2,3,\cdots,n\}$. If $a_i$ is the minimum element of set $A_i$ where $A_i\subset A$ such that $n(A_i)=3$, find the sum of all $a_i$ for all possible $A_i$ Number of subsets with least ...
67 views

### How do you deal with fractions in a binomial?

If I have something like this $$\binom{\frac{x}{k}}{\frac{y}{k}}$$ (where there are two fractions in a binomial but they have the same denominator) can I simplify this at all?
42 views

### What does this binomial sum equal?

I'm trying to evaluate this sum: $$\sum_{k=0}^n {n \choose k}{{2n+1}\choose k}$$ I thought I could work with generating functions of the two binomials. I know $$\sum_k\binom{n}k{}x^k=(1+x)^n$$ is the ...
86 views

### How to prove this equality about Eulerian numbers?

I want to prove the following equality where $A(k,m)$ is the Eulerian number : $$\forall k\ge0,\sum_{k=0}^{\infty}n^k x^k = \frac{\sum_{m=0}^{k-1}A(k,m)x^{m+1}}{(1-x)^{k+1}}$$ I previously proved ...
41 views

42 views

90 views

### How to derive this binomial identity?

I believe the following is an identity (I've tested with a few random $m$ and $n$ values, could be wrong though): $$\sum_{k= 0}^{\infty}{m \choose k}{n \choose k}k=n\binom{m+n-1}{m-1}$$ but I'm not ...
141 views

36 views

### Limit of sum of terms containing binomial coefficients

$$\lim_{n \to \infty} \sum_{k=0}^n \frac{n \choose k}{k2^n+n}$$ The result is $0$. The $n$ from the denominator can be ignored. If not for the $k$ at the denominator, the result would be $1$, but I ...
144 views

### Finite Double Sum $\sum_{j=0}^n\sum_{i=0}^j \binom {n+1}{j+1}\binom ni =2^{2n}$

The problem is given in a combinatorics class study sheet. I cannot prove, and actually I am not sure if there was a mistake in the question or not. I tried for a few small n's e.g. 1, 2 and it holds. ...
204 views

### Proving equality - a sum including binomial coefficient $\sum_{k=1}^{n}k{n \choose k}2^{n-k}=n3^{n-1}$

I want to prove the following equality: $$\displaystyle\sum_{k=1}^{n}k{n \choose k}2^{n-k}=n3^{n-1}$$ So I had an idea to use $((1+x)^n)'=n(1+x)^{n-1}$ So I could just use the binomial theorem and ...
40 views

### If $(1+2x)(1+x+x^2)^{n}=\sum_{r=0}^{2n+1} a_rx^r$ then find the required value

If $(1+2x)(1+x+x^2)^{n}=\sum_{r=0}^{2n+1} a_rx^r$ and $(1+x+x^2)^{s}=\sum_{r=0}^{2s} b_rx^r$, then value of $\frac{\sum_{s=0}^{n}\sum_{r=0}^{2s} b_r}{\sum_{r=0}^{2n+1} \frac{a_r}{r+1}}$ will be: (A) ...
114 views

### How many integer-sided right triangles are there whose sides are combinations?

How many integer-sided right triangles exist whose sides are combinations of the form $\displaystyle \binom{x}{2},\displaystyle \binom{y}{2},\displaystyle \binom{z}{2}$? Attempt: This seems like a ...
Number of terms in the expansion of $$\left(1+\frac{1}{x}+\frac{1}{x^2}\right)^n$$ $\bf{My\; Try::}$ We can write $$\left(1+\frac{1}{x}+\frac{1}{x^2}\right)^n=\frac{1}{x^{2n}}\left(1+x+x^2\right)^n... 4answers 411 views ### Finding coefficient of polynomial? The coefficient of x^{12} in (x^3 + x^4 + x^5 + x^6 + …)^3 is_______? Somewhere it explain as: The expression can be re-written as: (x^3 (1+ x + x^2 + x^3 + …))^3=x^9(1+(x+x^2+x^3))^3 ... 0answers 35 views ### Bounding the summation of binomial terms For 0<\theta<1, \theta'=(1-\epsilon)\theta, \epsilon< \theta, k\in\mathbb{N}, the problem is to tightly upper bound the following binomial summation:$$\sum_{i=\lceil \theta k \rceil}^k {...
I was learning generating functions, where faced following: $$(1-x)^{-n}=\sum_{i=0}^\infty\binom{n+i-1}{i}x^i$$ I didn't get how is this arrived. I tried out to prove this to me. But failed. ...