Intereting Posts

Star-shaped domain whose closure is not homeomorphic to $B^n$
Generalized Case: Three Consecutive Binomial Coefficients in AP
System of equations: $x^2+y=7, y^2+x=11$
When can one expect a classical solution of a PDE?
Mid-point convexity does not imply convexity
$f(\alpha x) = f(x)^{\beta}$ under different constraints
What kind of rings have exactly three ideals?
How unique (on non-unique) are U and V in Singular Value Decomposition (SVD)?
Learning Model Theory
If A is an infinite set and B is at most countable set, prove that A and $A \cup B$ have the same cardinality
Investigations about the trace form
Explicit construction of Haar measure on a profinite group
Can a vector subspace have a unique complement in absence of choice?
The ring of integers of the composite of two fields
Does rapid decay of Fourier coefficients imply smoothness?

Why is $\sum_{k=0}^n (_k^n) 2^k$ simplified as $3^n$ and not just $2^n$ ?

The answer uses the binomial theorem as a solution: let $a=1$ and let $b=2$. So $1+2 = 3$. But since $1+2 = 3$ why do we not use $1+1 = 2$?

- Find the population size that maximizes the probability that two random samples of size $20$ will have exactly $2$ members in common
- How many different possible permutations are there with k digits that add up to n?
- combinatorics circular arrangement problem
- Combinatorial proof of the identity ${n+1\choose k+1}={k\choose k}+{k+1\choose k}+…+{n-1\choose k}+{n\choose k}$
- Combinations problem involving a standard pack of $52$ playing cards and a $4$ sided die: Part 1
- Interpretation of a combinatorial identity
- prove that $\binom{n}{0}^2+\binom{n}{1}^2+\binom{n}{2}^2+\cdots+\binom{n}{n}^2=\binom{2n}{n}$
- A problem with my reasoning in a problem about combinations
- $4$ letter words taken from the letters of CONCENTRATIONS
- In how many ways can the letters of the word CHROMATIC be arranged?

It is $$\sum_{k=0}^n \binom{n}{k} 2^k \cdot 1^{n-k}=(1+2)^n$$

- Evaluate $\int_{0}^{+\infty }{\left( \frac{x}{{{\text{e}}^{x}}-{{\text{e}}^{-x}}}-\frac{1}{2} \right)\frac{1}{{{x}^{2}}}\text{d}x}$
- Prove that $\Gamma_I(\frac{M}{\Gamma_I(M)})=0$
- Interpretation of $p$-forms
- Integral of $\log(\sin(x))$ using contour integrals
- Discrete valuations of the rational numbers
- Every endomorphims is a linear combination of how many idempotents in infinite dimensions?
- Prove that $1+\frac{1}{3}+\frac{1\cdot 3}{3\cdot 6}+\frac{1\cdot 3\cdot 5}{3\cdot 6 \cdot 9}+…=\sqrt{3}$
- Differential of transposed matrices
- Card passing game, maximum length
- Decomposition of a prime number in a cyclotomic field
- Non-orthogonal projections summing to 1 in infinite-dimensional space
- Does $L^p$-convergence imply pointwise convergence for $C_0^\infty$ functions?
- For what $t$ does $\lim\limits_{n \to \infty} \frac{1}{n^t} \sum\limits_{k=1}^n \text{prime}(k)$ converge?
- Is the Bourbaki treatment of Set Theory outdated?
- Intersection of two isometric cubic graphs