How many different ways to write a positive integer $n$ as a sum of $m$ different positive integers? Different sequences are treated as distinct.
How many $3\times 3$ matrices of non-negative integers are there such that the sum of every row and every column equals $n$?
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 $x_1+x_2+\cdots + x_k =n$,
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.