# Tagged Questions

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

56 views

56 views

### How many $5$ card poker hands contain at least $1$ red and $1$ black card?

How many $5$ card poker hands contain at least $1$ red and $1$ black card? I used inclusion-exclusion to calculate my answer. The number of total poker card hands are:$$52\choose 5$$I have $26$ red ...
43 views

### Extending binomial identity $\sum\limits_{k=0}^n\frac{(-1)^k}{k+x}\binom{n}{k}\binom{n+k}{k}=0$ to $0<x<1$

I found in Matlab that $$\sum_{k=0}^n~\frac{(-1)^k}{k+x}\binom{n}{k}\binom{n+k}{k}=0$$ for $1\leq x< n$ only (I am about 95% sure of this since the sum is numerically unstable and cannot give ...
55 views

### Prove $\left(\dbinom nk \right)= \left(\dbinom{k+1}{n-1}\right)$ [closed]

I need to prove $\left(\!\dbinom nk \!\right)= \left(\!\dbinom{k+1}{n-1}\!\right)$ where the double parens denote multiset coefficients and $n,k$ are integers with $1 ≤ k≤ n$ using an algebraic proof. ...
29 views

### Proof binomial coefficient [closed]

I'm trying to prove the following: $$\binom{n + p}{k} = \sum_{j=0}^n \binom{n}{j} \cdot \binom{p}{k - j}$$ How do I do it? Induction? And can someone hint me at how to start?
65 views

32 views

47 views

### Vandermonde's Convolution special case.

I am not able to show this case of Vandermonde's Convolution without using induction. Can someone help me? $$\binom{n}{m} = \sum_{k=0}^{m} \binom{n-p}{m-k} \binom{p}{k}.$$ I thank now.
325 views

### Sum of sum of binomial coefficients $\sum_{j=1}^{n}{\sum_{k=0}^{j}{{n}\choose{k}}}$

I know there is no simple way to solve the sum: $$\sum_{k=0}^{j}{{n}\choose{k}}$$ But what about the equation: $$\sum_{j=1}^{n}{\sum_{k=0}^{j}{{n}\choose{k}}}$$ Are there any simplifications or ...
72 views

33 views

### Logic - Binomial Theorem

I could use some assistance with understanding this problem. I understand that there are ${n}\choose{k}$ is a representation of ${n}\choose{k}$ ways to choose k elements from a set of n elements, ...