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94 problems match “counting”. Clear

Odd reciprocals with terminating decimals ●●○○○ 20 ptsIf $\frac{a}{b}$ is the probability that the reciprocal of a randomly selected positive odd integer less than $2010$ gives a terminating decimal, with $a$ and $b$ relatively prime positive integers, find $a+b$ .
Greatest shared divisor of 90 and m ●●○○○ 20 ptsTwo numbers $90$ and $m$ share exactly three positive divisors. What is the greatest of these three common divisors?
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?
Distinct candy flavors by percent red ●●○○○ 20 ptsFour red candies and three green candies can be combined to make many different flavors. Two flavors are different if their percent red is different, so 3 red / 0 green is the same flavor as 2 red / 0 green, and 4 red / 2 green is the same flavor as 2 red / 1 green. If a flavor is made using some or all of the seven candies (at least one candy), how many different flavors are possible?
Removing an element to raise probability ●●○○○ 20 ptsTwo different numbers are randomly selected from the set $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$ . The probability that their sum is 12 would be greater if the number $n$ had first been removed from $S$ . What is the value of $n$ ?
Maximum leap years in 150 years ●●○○○ 20 ptsIn a calendar system in which leap years occur every four years, with no exceptions, what is the maximum possible number of leap years in a 150-year period?
Repeating decimals of n/30 ●●○○○ 20 ptsFor how many integers $n$ from $1$ to $29$ is the fraction $\frac{n}{30}$ a repeating decimal (that is, its decimal expansion does not terminate)?
Orders of breaking clay targets ●●○○○ 20 ptsEight clay targets are arranged in three hanging columns: two columns of three targets each and one column of two targets. A marksman breaks all the targets by these rules: 1. The marksman first chooses a column from which a target is to be broken. 2. The marksman must then break the lowest remaining target in the chosen column. If the rules are followed, in how many different orders can the eight targets be broken?
Subsets with a gap pattern ●●○○○ 20 ptsDetermine the number of subsets $S$ of $\{1,2,3,\ldots,10\}$ with the following property: there exist integers $a<b<c$ with $a \in S$ , $b \notin S$ , $c \in S$ .
Equal votes in a village election ●●○○○ 20 ptsA small village has $n$ people. During their yearly elections, every group of three people comes up to a stage and, as a group, casts a single vote for someone in the village to be the new leader. After every possible group of three people has voted, it turns out that every person in the village received exactly the same number of votes. If $10 \leq n \leq 100$ , how many possible values of $n$ are there?