Jackie and Phil have two fair coins and a third coin that comes up heads with probability $\frac47$. Jackie flips the three coins, and then Phil flips the three coins. Let $\frac {m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

How many different ways are there to express $20$ as the sum of $1$, $2$, and $5$? (All numbers must appear.)

There are $2$ white balls, $3$ red balls, and $1$ yellow ball in a jar. How many different ways are there to retrieve $3$ balls?

How many different $5$-digit numbers can be formed using $1$, $2$, $3$, and $4$ that satisfy the following conditions:

- the digit $1$ must appear either $2$ or $3$ times,
- the digit $2$ must appear even times,
- the digit $3$ must appear odd times, and
- the digit $4$ has no restriction

Show that $$\frac{1}{(1-x)^n}=\sum_{k=0}^{\infty}\binom{n-1+k}{n-1}x^k$$

Let $f(x)$ be the generating function for $a_0$, $a_1$, $a_2$, $\cdots$. Find the generating function for $$a_0, a_0 + a_1, a_0+a_1+a_2, \cdots$$

Show that $$\sqrt{1+x}=1+\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n\cdot 2^{2n-1}}\binom{2n-2}{n-1}x^n$$

Show that $$\frac{1}{\sqrt{1-4x}}=\sum_{n=0}^{\infty}\binom{2n}{n}x^n$$

Let $N$ be the value of the following expression. $$\sum_{k=0}^{n-1}\left(\binom{n}{0}+\binom{n}{1}+\cdots+\binom{n}{k}\right)\left(\binom{n}{k+1}+\binom{n}{k+2}+\cdots+\binom{n}{n}\right)$$

Show $$N=\frac{n}{2}\binom{2n}{n}$$

For every integer $n$ from $0$ to $6$, we have $3$ identical weights with weight $2^n$ grams. How many total ways are there to form a total weight of $263$ grams using only these weights?

How many $4$-digit integers are there whose sum of all digits equals $12$?

There are $10$ red, $10$ blue, and $10$ white balls. How many different ways are there to retrieve $16$ balls with all the three colors present.

Let $\alpha(n)$ be the number of ways to write a positive integer $n$ as the sum of $1$s and $2$s. Let $\beta(n)$ be the number of ways to write $n$ as a sum of several integers greater than $1$. Different orders are treated as different. Prove $\alpha(n)=\beta(n+2)$.

A deck of poker has three different colors each of which contains $10$ cards numbered from $1$ to $10$, respectively. In addition, there are two jokers both of which are numbered as $0$. A card with number $k$ is valued as $2^k$ points. How many different ways are there to draw several cards from this deck so that their total value equals $2004$?

Let $n$ be a positive integer. Find the number $a_n$ of polynomials $f(x)$ with coefficients in $\{0, 1, 2, 3\}$ such that $f(2)=n$.

Let $a_0$, $a_1$, $a_2$, $\cdots$ be an increasing sequence of non-negative integers such that every non-negative integer can be expressed uniquely in the form of $(a_i + 2a_j+4a_k)$ where $i$, $j$, and $k$ are not necessarily distinct. Determine $a_{1998}$.

Let $\mathbb{N}$ be the set containing all positive integers. Is it possible to partition $\mathbb{N}$ to more than one but still a finite number of arithmetic sequences with no two having the same common difference?

Let $p$ be an odd prime number. Find the number of subsets $\mathbb{A}$ of the set $\{1, 2, \cdots, 2p\}$ such that $\mathbb{A}$ has exactly $p$ elements and the sum of all elements in $\mathbb{A}$ is divisible by $p$.

Let $a$, $b$, $p$, and $q$ be fixed positive integers. If an $a\times b$ grid can be tiled using some $1\times p$ and $q\times 1$ pieces, show that either $a$ is divisible by $p$ or $b$ is divisible by $q$. Here, a $1\times k$ and $k\times 1$ grids are treated as different.

Show that $$\frac{1}{1-ax}=\sum_{k=0}^{\infty}(ax)^n$$

Show that $$\frac{1}{1-x}=1+x+x^2+x^3+x^4 + \cdots$$

Show that $$\frac{1}{1+x}=\sum_{k=0}^{\infty}(-1)^nx^n$$

(right shifting) Find the generating function for the sequence: $$\underbrace{0, 0, \cdots, 0}_{k}, 1, 1, 1, 1, \cdots$$

Show that $$\frac{1}{1-x^2}=\sum_{k=0}^{\infty}x^{2k}$$

Show that $$\frac{1}{(1-ax)^n}=\sum_{k=0}^{\infty}a^k\binom{n-k}{n-1}x^k$$