We have four cards of different sizes. We are to paint each card either red, black, blue or yellow. However, we may paint more than one card with the same color. (1) There are a total of $\mathbf { N O P }$ ways of painting the cards. (2) There are $\mathbf{QR}$ ways of painting them using all four colors. (3) There are $\mathbf{ST}$ ways of painting two cards red, one card black, and one card blue. (4) There are $\mathbf{UVW}$ ways of painting the four cards using three colors. (5) There are $\mathbf { X Y }$ ways of painting them using two colors.
We have four cards of different sizes. We are to paint each card either red, black, blue or yellow. However, we may paint more than one card with the same color. (1) There are a total of $\mathbf { N O P }$ ways of painting the cards. (2) There are $\mathbf{QR}$ ways of painting them using all four colors. (3) There are $\mathbf{ST}$ ways of painting two cards red, one card black, and one card blue. (4) There are $\mathbf{UVW}$ ways of painting the four cards using three colors. (5) There are $\mathbf{XY}$ ways of painting them using two colors.
A teacher requires the class monitor to distribute review sheets for Chinese, English, Mathematics, Social Studies, and Science—5 subjects in total—over Monday, Tuesday, Wednesday, and Thursday of next week. At least one subject's sheet must be distributed each day for students to take home for practice and submit the next day. Since there are Chinese and English classes on Tuesday, the Chinese teacher requires that the Chinese sheet must be distributed on Monday for review; and the English teacher, having assigned other work that day, requires that the English sheet not be distributed on Tuesday. According to these requirements, the class monitor has ways to arrange the distribution.
A washing machine cycle must select one from 5 different fabric types (1, 2, 3, 4, 5), paired with one of 4 different modes (A, B, C, D), and there are 3 additional functions (A, B, C) that can be freely chosen to enable or disable. However, ``fabric type 1'' cannot be used simultaneously with additional function ``A''. For example, ``fabric type 2'' paired with ``mode A'', with both ``A'' and ``B'' additional functions enabled is a valid cycle; but ``fabric type 1'' paired with ``mode A'', with both ``A'' and ``B'' additional functions enabled is not a valid cycle. Based on the above, this washing machine has how many valid cycles?
A company hires 8 new employees, including 2 translators, 3 engineers, and 3 assistants. These 8 people are assigned to research and testing departments, with 4 people assigned to each department. Each department must include 1 translator and at least 1 engineer. There are (15--1)(15--2) ways to make such assignments.
There are $n$ children queuing in a line. You have $m$ candies and will begin handing out 1 or 2 candies to each child, starting from the first child in the line. You hand out the candies until reaching the end of the line or until there are no candies left. Answer the following questions. Note that $n$ and $m$ are positive integers. I. Show the number of distribution patterns of candies if $n = m = 4$. II. Show the number of distribution patterns of candies if $m \geq 2 n$. III. Define $X _ { m }$ as the number of distribution patterns of candies if $n \geq m$. Show the recurrence formula satisfied by $X _ { m }$. IV. Obtain $X _ { m }$ using the recurrence formula in Question III. V. Consider the situation where the number of children is larger than the number of candies. Define $P ( m )$ as the ratio of the number of distribution patterns (where the distribution finishes by giving 2 candies) to the total number of distribution patterns. $P ( m )$ converges as $m$ increases. Compute the convergence value. VI. Consider the situation where $m \geq 2 n$. The following rules are added to the way of handing out the candies: For the first child in the line, the probability of receiving 1 candy is $1 / 2$ and the probability of receiving 2 candies is $1 / 2$. If a child receives 1 candy, the probability of the next child receiving 1 candy is $1 / 2$ and the probability of receiving 2 candies is $1 / 2$. If a child receives 2 candies, the probability of the next child receiving 1 candy is $3 / 4$ and the probability of receiving 2 candies is $1 / 4$. Compute the probability that the $n$-th child in the line receives 2 candies.
An airline has a total of 8 cabin crew members with different work experience for one morning and one evening flight to be performed. Each of these employees will be on only one team, and two four-person flight teams will be formed from these employees such that the three most experienced employees are not on the same team. Accordingly, in how many different ways can the morning and evening flight teams be formed? A) 48 B) 54 C) 56 D) 60 E) 64
Regarding the sets $A, B, C, K$ and $L$, $$K = A \times B$$ $$L = B \times C$$ are given. Given that $K \cup L = \{(1,2), (1,3), (2,2), (3,2), (3,3)\}$, which of the following is the set $K \cap L$? A) $\{(1,2)\}$ B) $\{(1,3)\}$ C) $\{(2,2)\}$ D) $\{(3,2)\}$ E) $\{(3,3)\}$