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Arrangements of letters

Solve the first problem 17 problems

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1. Arrangements of the letters of TOOT ●○○○○ 10 ptsDetermine the number of distinct ways to arrange the letters of the word TOOT.
2. Arrangements of six letters ●○○○○ 10 ptsHow many permutations can be made from the set $\{A, E, I, O, U, Y\}$ using all six letters?
3. Three-letter words from LOVE ●○○○○ 10 ptsHow many 3-letter arrangements (words need not be meaningful) can we make using the letters of the word LOVE, each letter used at most once?
4. 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?
5. Passwords with equal lowercase counts ●●○○○ 20 ptsComputer passwords are between 6 and 8 characters long. Each character is an upper case letter, a lower case letter or a digit. Each password has at least one upper case letter, at least one lower case letter and at most one digit. Let $S$ be a set of such passwords. What is the minimum size of $S$ that guarantees at least two passwords in $S$ have the same number of lower case letters?
6. 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?
7. Almost palindromic letter arrangements ●●●○○ 50 ptsCall a string of letters $S$ an almost palindrome if $S$ and the reverse of $S$ differ in exactly two places. Find the number of ways to order the letters in $HMMTTHEMETEAM$ to get an almost palindrome.
8. 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?
9. Six-letter words using all letters ●●●○○ 50 ptsSix-letter "words" are formed using the letters A, B, C and D, with letters allowed to repeat. In how many of them does each letter appear at least once?
10. 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 $z$?

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