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8 problems match “counting with restrictions”. Clear

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1. Strings with distinctness constraints ●●○○○ 20 ptsAlec wishes to construct a string of 6 letters using the letters A, C, G, and N, such that: - the first three letters are pairwise distinct, and so are the last three letters; - the first, second, fourth, and fifth letters are pairwise distinct. In how many ways can he construct the string?
2. Gifts to five children with restrictions ●●○○○ 20 ptsFour types of present are available: a candy, a paperback, a dolly, and a Mercedes. Five children, Jackie, Tito, Jermaine, Marlon and Michael, each receive exactly one present. Michael and Jermaine cannot receive the same type of present, Tito and Michael cannot receive the same type of present, and Tito cannot receive a paperback. In how many ways can the presents be given?
3. Seven-letter good words ●●○○○ 20 ptsDefine a good word as a sequence of letters that consists only of the letters $A$, $B$, and $C$ (some of these letters may not appear in the sequence), and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, and $C$ is never immediately followed by $A$. How many seven-letter good words are there?
4. Arranging letters of COMPUTE ●●●○○ 50 ptsHow many ways are there to arrange the letters of the word COMPUTE so that the vowels are in alphabetical order, or the consonants are in alphabetical order, or at least two vowels are next to each other?
5. Blue rectangles on a blackboard ●●●○○ 50 ptsBlue and red rectangles are drawn on a blackboard. Exactly 7 of the rectangles are squares. There are 3 more red rectangles than blue squares. There are 2 more red squares than blue rectangles. How many blue rectangles are there?
6. Minimum red balloons in a string ●●●●○ 100 ptsDavid is making a string of red, white, and blue balloons by these rules: - no red balloon is adjacent to another red balloon; - white balloons appear in groups of exactly two, and groups of white balloons are separated by at least two non-white balloons; - blue balloons appear in groups of exactly three, and groups of blue balloons are separated by at least three non-blue balloons. If David uses more than 600 balloons, find the smallest number of red balloons he can use.
7. Counting case-switching letter sequences ●●●●○ 100 ptsBob writes a sequence of letters of the alphabet, each of which is either uppercase or lowercase, under these rules. If he has just written an uppercase letter, he either writes the same letter in lowercase next, or the next letter of the alphabet in uppercase. If he has just written a lowercase letter, he either writes the same letter in uppercase next, or the preceding letter of the alphabet in lowercase. For example, $a\,A\,a\,A\,B\,C\,D\,d\,c\,b\,B\,C$ is such a sequence. How many sequences of 32 letters can he write that start at lowercase $a$ and end at lowercase $z$?
8. Maximum smallest ball in bowl Jia ●●●●○ 100 ptsPlace the 27 balls numbered $1$ to $27$ into three bowls, Jia, Yi and Bing, so that the average ball number in bowl Jia is $15$, in bowl Yi is $3$, and in bowl Bing is $18$, and each bowl contains at least $4$ balls. Find the maximum possible value of the smallest ball number in bowl Jia.