1. Suppose we randomly choose nonnegative integers $x_1, \, x_2, \, x_3, \, \text{and } x_4$ that solve the equation $x_1 + x_2 + x_3 + x_4 = 10.$ We assume that each solution has an equal probability of being chosen. Given that at least one of $x_1$ and $x_2$ is equal to $2,$ what is the probability that $x_2 = 2?$ (15%)
參考解答：$\dfrac{45}{83}$
2. Suppose that Mark selects a ball by first picking one of two boxes at random and then selecting a ball from this box at random. The first box contain 5 red balls and 4 blue balls, and the second box contain 3 red balls and 6 blue balls. What is the probability that Mark picked a ball from the second box if he has selected a red ball? (15%)
參考解答：$\dfrac{3}{8}$
3. Solve the following recurrence relation:
$$3a_n – 6a_{n-1} – 3a_{n-2} + 6a_{n-3} = 0$$
with $a_0 = 1, \, a_1 = 0, \, a_2 = 7.$ (10%)
參考解答：$a_n = \dfrac{-5}{2} + \dfrac{3}{2}(-1)^n + 2^{n+1}, \, n \geq 0.$
4. Find the set of all solutions $x$ to the system of congruences: (10%)
$$x \equiv 4(\text{mod }5) \quad \text{and} \quad x \equiv 5(\text{mod }15)$$
參考解答：The system has no solution.
試題(pdf)：連結
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