Search problems

94 problems match “counting”. Clear

Pairs of unit complex numbers ●●○○○ 20 ptsFind the number of pairs $(z_1,z_2)$ of complex numbers such that $z_1 z_2$ is pure imaginary, $\frac{z_1}{z_2}$ is real, and $|z_1| = |z_2| = 1$ .
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?
Permutations with order dividing twelve ●●○○○ 20 ptsLet $A=\{1,2,\dots,6\}$ and let $f: A \to A$ be a one-to-one mapping. Define $f^{(2)}(x)=f(f(x))$ and $f^{(n+1)}(x)=f(f^{(n)}(x))$ for $n \ge 1$ . Find the number of mappings $f$ such that $f^{(12)}(x)=x$ for every $x \in A$ .
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?
Minimum matches to find a winner ●●○○○ 20 ptsIn a badminton knockout-style contest with 51 players, a player is eliminated when he loses two matches. What is the minimum number of matches needed to determine the winner? Assume every match has exactly one winner and one loser, and that the schedule can be chosen freely.
Subsets contained in neither of two sets ●●○○○ 20 ptsFind the number of subsets of $\{1, 2, 3, 4, 5, 6, 7, 8\}$ that are subsets of neither $\{1, 2, 3, 4, 5\}$ nor $\{4, 5, 6, 7, 8\}$ .
Expected consecutive pairs in a subset ●●○○○ 20 ptsA 25-element subset is chosen uniformly at random from $\{1,2,\ldots,100\}$ . Find the expected number of pairs $(i,i+1)$ of consecutive integers for which both $i$ and $i+1$ lie in the subset.
Linear function avoiding the second quadrant ●●○○○ 20 ptsTwo different numbers are chosen from $-2,-1,1,2,3$ , the first as $k$ and the second as $b$ , to form the linear function $y=kx+b$ . Find the probability that the resulting line does not pass through the second quadrant. Give your answer as a decimal.
Sizes of two sets from subset counts ●●○○○ 20 ptsLet $A$ and $B$ be two finite sets. The number of subsets of $A$ minus the number of subsets of $B$ is $112$ . Find the number of elements in $A$ and in $B$ .
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?