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94 problems match “counting”. Clear

Terminating decimals with nonzero hundredths digit ●●●○○ 50 ptsFor how many positive integers $n$ does $\frac{1}{n}$ yield a terminating decimal with a non-zero hundredths digit?
Placing plants under colored lamps ●●●○○ 50 ptsRachel has two identical basil plants and one aloe plant. She has two identical white lamps and two identical red lamps. Each plant is placed under exactly one lamp, and more than one plant may share a lamp. How many ways are there for Rachel to place her plants under her lamps?
Exactly two friends visiting Daphne ●●●○○ 50 ptsDaphne is visited periodically by her three best friends, Alice, Beatrix and Claire. Alice visits every third day, Beatrix every fourth day, and Claire every fifth day. All three friends visited Daphne yesterday. On how many days of the next 365-day period will exactly two friends visit her?
Each player gets an ace ●●●○○ 50 ptsA standard 52-card deck is shuffled and 2 cards are dealt to each of 4 players. Find the probability that every player receives at least one ace. Give your answer as a decimal rounded to eight decimal places.
Integers with distinct remainders ●●●○○ 50 ptsHow many positive integers at most 420 leave different remainders when divided by each of 5, 6, and 7?
One ace per suit in pinochle ●●●○○ 50 ptsA player is dealt a 20-card hand from a double-deck pinochle deck of 80 cards, which contains 4 aces of each suit. Find the probability that the hand contains exactly 1 ace of each of the four suits. Round to four decimal places.
Counting lines with acute inclination ●●●○○ 50 ptsLet the line $ax+by+c=0$ have $a$ , $b$ , $c$ as three different elements of the set $\{-3,-2,-1,0,1,2,3\}$ , and suppose the inclination angle of the line is acute. Find the number of distinct such lines.
Counting super keys from dependencies ●●●○○ 50 ptsA relation $R$ has attributes $A, B, C, D, E, F, G, H$ and the functional dependencies $CH \to G$ , $A \to BC$ , $B \to CFH$ , $E \to A$ , $F \to EG$ . How many super keys does $R$ have? A super key is a set of attributes whose closure contains all attributes of $R$ .
Reachable values on a blackboard ●●●●○ 100 ptsSvitlana writes the number 147 on a blackboard. Then, at any point, if the number on the blackboard is $n$ , she can perform one of the following three operations: if $n$ is even, she can replace $n$ with $\frac{n}{2}$ ; if $n$ is odd, she can replace $n$ with $\frac{n+255}{2}$ ; and if $n \geq 64$ , she can replace $n$ with $n-64$ . Compute the number of possible values that Svitlana can obtain by doing zero or more operations.
Minimum number of 7-subsets covering all triples ●●●●○ 100 ptsLet $S=\{1,2,\ldots,15\}$ . Choose $n$ subsets $A_1,A_2,\ldots,A_n$ of $S$ such that: (i) $|A_i|=7$ for each $i$ ; (ii) $|A_i\cap A_j|\leq 3$ for all $1\leq i<j\leq n$ ; (iii) for every 3-element subset $M$ of $S$ , there is some $k$ with $M\subset A_k$ . Find the minimum possible value of $n$ .