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17 problems match “permutations”. Clear

Last passenger in assigned seat ●●●○○ 50 ptsOne hundred passengers board a full flight in order, each with an assigned seat. The first passenger loses their boarding pass and picks a seat at random. Each later passenger sits in their assigned seat if it is free, and otherwise picks a seat at random from the remaining free seats. What is the probability that the last passenger sits in their assigned seat? Give the answer as a decimal.
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.
Expected count of numbers exceeding a neighbor ●●●○○ 50 ptsThe numbers $1$ to $10$ are written in a circle in a random order. Find the expected number of numbers that are at least $2$ larger than at least one of their two adjacent numbers. Round to two decimal places.
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?
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?
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.
Birthdays that are equal or adjacent ●●●●○ 100 ptsAssume 23 people have independent birthdays, each equally likely to be any of 365 days (February 29 is not considered), and January 1 and December 31 are considered adjacent. Find the probability that at least two people share the same birthday or have adjacent birthdays. Round to seven decimal places.