Intereting Posts

Why is the determinant equal to the index?
Prerequisite to start learning Topological K-Theory
Prerequisite for Petersen's Riemannian Geometry
How find this $\lim_{n\to\infty}n^2\left(\frac{1^k+2^k+\cdots+n^k}{n^{k+1}}-\frac{1}{k+1}-\frac{1}{2n}\right)$
Time until a consecutive sequence of ones in a random bit sequence
Books for combinatorial thinking
Integration (Fourier transform)
$\mathbb Z$ is not finitely generated?
In a metric space, is every open set the countable union of closed sets?
simplify boolean expression: xy + xy'z + x'yz'
$4$-digit positive integers that does not contain the digits $3$ and $4$ plus other properties
Is the space of $G$-maps $G/H \to X$ naturally homeomorphic to $X^H$?
Question concerning tensors
How to show that $\frac{\pi}{5}\leq\int_0^1 x^x\,dx\leq\frac{\pi}{4}$
Why this space is homeomorphic to the plane?

It’s easy to prove $\sum_{i=0}^n {{i}\choose{ k}} = {{n+1}\choose{ k+1}}$ by induction with ${{n}\choose{ k}} = {{n-1}\choose{ k-1}} + {{n}\choose{ k-1}}$, but what is its combinatorial proof?

- Is this relation reflexive?
- Number of ways to put $n$ unlabeled balls in $k$ bins with a max of $m$ balls in each bin
- Let $A$ be any uncountable set, and let $B$ be a countable subset of $A$. Prove that the cardinality of $A = A - B $
- Prove $7$ divides $13^n- 6^n$ for any positive integer
- How many passwords can be formed from this maze?
- Recurrence relation (linear, second-order, constant coefficients)
- Need a counter example for cycle in a graph
- Mathematical induction prove that 9 divides $n^3 + (n+1)^3 + (n+2)^3$ .
- Mathematical Induction for Recurrence Relation
- Moscow puzzle. Number lattice and number rearrangement. Quicker solution?

How many ways can we pick a subset of size $k+1$ from $n+1$? Let’s say the final element of our selection appears at index $i$. Then we need to pick $k$ elements from the previous $i-1$.

So the total number of ways is $$\sum_{i=1}^{n+1} \binom{i-1}{k} = \sum_{i=0}^n \binom{i}{k}$$

- What is the time complexity of Euclid's Algorithm (Upper bound,Lower Bound and Average)?
- Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)
- A.P. terms in a Quadratic equation.
- combining conditional probabilities
- Categorical Pasting Lemma
- Number of surjective functions from A to B
- How many different words can be formed using all the letters of the word GOOGOLPLEX?
- lacunary sum of binomial coefficients
- Try not to get the last piece, but with a twist!
- Most succinct proof of the uniqueness and existence of the Levi-Civita connection.
- How to construct a function from a pair of possibly empty sets?
- Topological properties of symmetric positive definite matrices
- How much does symbolic integration mean to mathematics?
- Unpacking the Diagonal Lemma
- Calculate the lim $\lim_{x\to 1} \frac{ \sqrt{x} – 1}{\sqrt{x} – 1}$