Intereting Posts

Conditional probability containing two random variables
Locally Constant Functions on Connected Spaces are Constant
Inequality for expected value
Summing the series $(-1)^k \frac{(2k)!!}{(2k+1)!!} a^{2k+1}$
Can any meromorphic function be represented as a product of zeroes and poles?
Is the curl of every non-conservative vector field nonzero at some point?
Orientation and simplicial homology
Proving $\text{Li}_3\left(-\frac{1}{3}\right)-2 \text{Li}_3\left(\frac{1}{3}\right)= -\frac{\log^33}{6}+\frac{\pi^2}{6}\log 3-\frac{13\zeta(3)}{6}$?
For which conditions on countable sets does continuity implies uniform continuity
Reducibility of $P(X^2)$
Stuff which squares to $-1$ in the quaternions, thinking geometrically.
How should I understand the $\sigma$-algebra in Kolmogorov's zero-one law?
The way into logic, Gödel and Turing
What's the geometrical interpretation of the magnitude of gradient generally?
Primality of number 1

In a book, I found that the sum of combinations: $\binom{n}{k} + \binom{n}{k+1} +\cdots+ \binom{n}{n}$, where *k* starts from 0, equals $2^n$. It is possible to express this statement via sum: $2 + \sum_{k=1}^{n-1}{n!\over k!(n-k)!} = 2^n$, using binomial theorem. I tried several small n values, such as 4, 6 and others, the statement looks correct. But I can’t find a regularity, respectively I can’t understand, **how to prove the truth of the statement?**

- Mathematically representing combinations with integers uniquely?
- Probability that $7^m+7^n$ is divisible by $5$
- $m\times n$ matrix with an even number of 1s in each row and column
- How many $N$ digits binary numbers can be formed where $0$ is not repeated
- Coloring a cube with 4 colors
- number of pairs formed from $2n$ people sitting in a circle
- Hypersurfaces containing given lines
- How many different ways can a number N be expressed as a sum of K different positive integers?
- A simple permutation question - discrete math
- Why am I under-counting when calculating the probability of a full house?

$$(x+y)^n=\sum_{k=0}^n \binom{n}{k} x^k y^{n-k}$$

For $x=1$ and $y=1$:

$$2^n=\sum_{k=0}^n \binom{n}{k}=\binom{n}{0}+\binom{n}{1}+ \dots +\binom{n}{n-1}+\binom{n}{n}$$

- Showing $\lim_{n\rightarrow\infty}\sqrt{n^3+n^2}-\sqrt{n^3+1}\rightarrow\frac{1}{3}$
- Does there exist a linearly independent and dense subset?
- What is a support function: $\sup_{z \in K} \langle z, x \rangle$?
- $7$ points inside a circle at equal distances
- Conjecture about integral $\int_0^1 K\left(\sqrt{\vphantom1x}\right)\,K\left(\sqrt{1-x}\right)\,x^ndx$
- Prove or disprove: For $2\times 2$ matrices $A$ and $B$, if $(AB)^2=0$, then $(BA)^2=0$.
- Is $\log(z^2)=2\log(z)$ if $\text{Log}(z_1 z_2)\ne \text{Log}(z_1)+\text{Log}(z_2)$?
- If all Subgroups are Cyclic, is group Cylic?
- How to find the maximum diagonal length inside a dodecahedron?
- union of two independent probabilistic event
- Maximal real subfield of $\mathbb{Q}(\zeta )$
- How is $L_{2}$ Minkowski norm different from $L^{2}$ norm?
- $(a,b)=ab$ in non factorial monoids
- Normal subgroups of p-groups
- Accumulation points of $\{ \sqrt{n} – \sqrt{m}: m,n \in \mathbb{N} \}$