Joe and Mary flip a coin ($n+1$) and $n$ times, respectively. What is the probability that Joe gets more heads than Mary does?

$\textbf{Seat on a Flight}$

There are $100$ airline passengers waiting in line to board a $100$-seat plane. For convenience, let the $n^{th}$ passenger in line hold a ticket for the $n^{th}$ seat. For some reasons, the first passenger decides to pick a random seat instead of his assigned seat (it is still possible that he or she picks the $1^{st}$ seat). Everybody will sit on his or her assigned seat unless this seat is occupied. In the latter case, that passenger will pick a random seat for himself or herself. Find the probability that the last passenger will sit on his or her assigned seat.

$\textbf{Boys v.s. Girls}$

In a remote town, people generally prefer boys over girls. Therefore, every married couple will continue giving birth to a baby until they have a son. Assuming there is fifty-fifty chance for a couple to give birth to a boy or a girl, what is the ratio of boys to girls in this town over many years?

$\textbf{Number of Routes}$

The shortest route from point $A$ to $B$ takes $10$ steps. How many such routes are there that do not pass point $C$?

A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square?

Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7$. Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?

Explain why we cannot apply the cut-the-rope technique to count the non-negative integer solutions to the equation $$x_1 + x_2 + \cdots + x_k = n$$

For example, can we allow two cuts in the same interval thus to model one of the $x_i$ is zero?

Let $n \ge k$ are two positive integers. Given function $x_1+x_2+\cdots + x_k =n$,

1. Find the number of positive integer solutions to this equation.
2. Find the number of non-negative integer solutions to this equation.
3. Explain the relation between these two cases. i.e. is it possible to derive (2) from (1), and vice versa?

Explain why the count of positive / non-negative integer solutions to the equation $x_1 + x_2 + \cdots + x_k=n$ is equivalent to the case of putting $n$ indistinguishable balls into $k$ distinguishable boxes.

Randomly draw a card twice with replacement from $1$ to $10$, inclusive. What is the probability that the product of these two cards is a multiple of $7$?

How many even $4$- digit integers are there whose digits are distinct?

Derive the permutation formula $P_n^n=n\times (n-1)\times\cdots\times 2\times 1$ using the recursion method.