A child builds towers using identically shaped cubes of different colors. How many different towers with a height of $8$ cubes can the child build with $2$ red cubes, $3$ blue cubes, and $4$ green cubes? (One cube will be left out.)

Equally divide each side of a triangle into $n$ parts and then connect these points to draw lines which are parallel to one of the triangle's sides. Find the number of parallelograms created by these lines.

Dividing a circle into $n \ge 2$ sectors and coloring these sectors using $m\ge 2$ different colors. If no adjacent sectors can be colored the same, how many different color schemes are there?

Find the number of non-negative integer solutions to the equation $$2x_1+x_2+x_3+\cdots+x_9+x_{10}=3$$

Let $\mathbb{S}=\{1,\ 2,\ 3,\ \cdots,\ n\}$ and positive integer $m$ satisfying $n + 1\ge 2m$. Find the number of subsets of $\mathbb{S}$ which has $m$ elements and no two elements are consecutive.

How many ordered integers $(x_1,\ x_2,\ x_3,\ x_4)$ are there such that $0 < x_1 \le x_2\le x_3\le x_4 < 7$?

How many different ways to write a positive integer $n$ as a sum of $m$ different positive integers? Different sequences are treated as distinct.

Let $\mathbb{S} =\{a_1,\ a_2,\ \cdots,\ a_n\}$ be a permutation of $\{1,\ 2,\ \cdots,\ n\}$ which satisfies the condition that for every $a_i$, $(i=1$, $2$, $\cdots$, $n)$, there exists an $a_j$ where $i< j \le n$ such that $a_j=a_i+1$ or $a_j=a_i-1$. Find the number of such $\mathbb{S}$.

Let $\mathbb{S}=\{a_1,\ a_2,\ \cdots,\ a_n\}$ where every element $a_i\in\{1,\ 2,\ \cdots,\ k\}$. Find the number of $\mathbb{S}$ which has an even number of $1$s.

In the Banana Country, only Mr Decent always tells the truth and only Mr Joke always tells lies. Everyone else has a probability of $p$ to tell a lie. One day, Mr Decent has decided to run for the President and told his decision to the first person who in turn told this to the second person. The second person then told this to the third person, and so on, till the $n^{th}$ person who told this news to Mr Joke. No one has been told this news twice in this process. Finally, Mr Joke announced Mr Decent's decision to everyone. What is the probability that Mr Joke's statement agrees with Mr Decent's intention?

A permutation $\{x_1,\ x_2,\ \cdots,\ x_{2n}\}$ of the set $\{1,\ 2,\ \cdots,\ 2n\}$, where $n$ is a positive integer, is said to have property $P$ if $\mid x_i − x_{i+1}\mid = n$ for at least one $i$ in $\{1,\ 2,\ \cdots,\ 2n − 1\}$. Show that, for each $n$, there are more permutations with property $P$ than without.

Let $S_n$ be the number of non-congruent triangles whose sides' lengths are all integers and circumferences equals $n$. Show that $$S_{2n-1}-S_{2n} = \left\lfloor\frac{n}{6}\right\rfloor\quad\text{or}\quad\left\lfloor\frac{n}{6}\right\rfloor +1$$

where $\lfloor{x}\rfloor$ returns the largest integer not exceeding the real number $x$.

How many $3\times 3$ matrices of non-negative integers are there such that the sum of every row and every column equals $n$?

Let $n$, $m$ and $k$ be three positive integers satisfying $m(k-1) < n$. Find the number of ways to select $k$ items from $\{1,\ 2,\ \cdots,\ n\}$ for form a strict increasing sequence and the difference between adjacent terms is no more than $m$.

As shown, an isosceles trapezoid is obtained by removing the top part of an equilateral triangle. The lengths of its two bases are $a$ and $b$, respectively, which are both integers. Both bases and sides are equally divided into unit-length parts. Their ending points are then connected to create several segments which are parallel to either two bases or one side. Find the number of equilateral triangles in this diagram.

Find the number of ways to divide a convex $n$-sided polygon into $(n-2)$ triangles using non-intersecting diagonals.

Given two distinct values $b_1$ and $b_2$, their product can be written in two ways: $b_1\times b_2$ and $b_2\times b_1$. Given three distinct values $b_1$, $b_2$, and $b_3$, their products can be expressed in $12$ ways: $b_1\times(b_2\times b_3)$, $(b_1\times b_2)\times b_3$, $b_1\times(b_3\times b_2)$, $(b_1\times b_3)\times b_2$, $b_2\times(b_3\times b_1)$, $(b_2\times b_3)\times b_1$, $b_2\times(b_1\times b_3)$, $(b_2\times b_1)\times b_3$, $b_3\times(b_1\times b_2)$, $(b_3\times b_1)\times b_2$, $b_3\times(b_2\times b_1)$, and $(b_3\times b_2)\times b_1$. Please note that in this definition, orders matter and parentheses etc cannot be simplified. The question is how many different ways to express the product of $n$ distinct values?

Let $x_i\in\{+1,\ -1\}$, $i=1,\ 2,\ \cdots,\ 2n$. If their sum equals $0$ and the following inequality holds for any positive integer $k$ satisfying $1\le k < 2n$: $$x_1+x_2+\cdots + x_k\ge 0$$

Find the number of possible ordered sequence $\{x_1,\ x_2,\ \cdots,\ x_{2n}\}$.

(Hanoi Tower) There are $3$ identical rods labeled as $A$, $B$, $C$; and $n$ disks of different sizes which can be slide onto any of these three rods. Initially, the $n$ disks are stacked in ascending order of their sizes on $A$. What is the minimal number of moves in order to transfer all the disks to $B$ providing that each move can only transfer one disk to another rod's topmost position and at no time, a bigger disk can be placed on top of a smaller one.

Find the total number of sequences of length $n$ containing only letters $A$ and $B$ such that no two $A$s are next to each other. For example, for $n = 2$, there are $3$ possible sequences: $AB$, $BA$, and $BB$.

How many distinct permutations of the letters of the word REDDER are there that do not contain a palindromic substring of length at least two? (A substring is a contiguous block of letters that is part of the string. A string is palindromic if it is the same when read backwards.)

Contessa is taking a random lattice walk in the plane, starting at $(1, 1)$. (In a random lattice walk, one moves up, down, left, or right $1$ unit with equal probability at each step.) If she lands on a point of the form $(6m, 6n)$ for $m$, $n\in\mathbb{Z}$, she ascends to heaven, but if she lands on a point of the form $(6m+ 3, 6n+ 3)$ for $m,\ n\in\mathbb{Z}$, she descends to hell. What is the probability that she ascends to heaven? 

In an election for the Peer Pressure High School student council president, there are $2019$ voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice’s boyfriend Bob votes for Alice as well. Then one by one, each of the remaining $2016$ voters votes for a candidate randomly, with probabilities proportional to the current number of the respective candidate’s votes. For example, the first undecided voter David has a $2/3$ probability of voting for Alice and a $1/3$ probability of voting for Celia. What is the probability that Alice wins the election (by having more votes than Celia)?

Solve the recursion $$a_n=\sum^{n-1}_{k=0}a_{k}a_{n-k-1}=a_0a_{n-1}+a_1a_{n-2}+\cdots+a_{n-1}a_0$$

where $a_0=a_1=1$.