Practice (68)
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Avi and Hari agree to meet at their favorite restaurant between 5:00 p.m. and 6:00 p.m. They have agreed that the person who arrives first will wait for the other only 15 minutes before leaving. What is the probability that the two of them will actually meet at the restaurant, assuming that the arrival times are random within the hour? Express your answer as a common fraction.
What is the mean of all possible positive three-digit integers in which no digit is repeated and all digits are prime? Express your answer as a decimal to the nearest hundredth.
A circular spinner has seven sections of equal size, each of which is colored either red or blue. Two colorings are considered the same if one can be rotated to yield the other. In how many ways can the spinner be colored?
What is the probability that a randomly selected integer from $1$ to $81$, inclusive, is equal to the product of two one-digit numbers?
How many diagonals does a convex octagon have?
Jean is twice as likely to make a free throw as she is to miss it. What is the probability that she will miss $3$ times in a row?
We have a set of numbers {1, 2, 3, 4, 5} and we take products of three different numbers. We must find now many pairs of relatively prime numbers there are.
In how many ways can 6 different gifts be given to five different children with each child receiving at least one gift and each gift being given to exactly one child?
For how many two-element subsets {$a,b$} of the set {$1, 2, 3, \cdots, 36$} is the product of $ab$ a perfect square?
In one roll of four standard six-sided dice, what is the probability of rolling exactly three different numbers?
In how many ways can $18$ be written as the sum of four distinct positive integers?
In a stack of six cards, each card is labeled with a different integer from $0$ to $5$. Two cards are selected at random without replacement. So what is the probability that their sum will be $3$?
A bag contains ten each of red and yellow balls. The balls of each color are numbered from 1 to 10. If two balls are drawn at random, without replacement, then what is the probability that the yellow ball numbered 3 is drawn followed by a red ball?
In the figure shown, the diagonals of a square are drawn and then two additional segments from each vertex to a diagonal. How many triangles are in the figure?
A $5 \times 5 \times 5$ cube is painted on 5 of its 6 faces. It is then cut into 125 unit cubes. One unit cube is randomly selected and rolled. We are asked to find the probability that the top face of the cube that is rolled is painted.
If the letters of the word ELEMENT are randomly arranged, what is the probability that the three E's are consecutive?
Four boys and four girls line up in a random order. What is the probability that both the first and last person in line is a girl?
Meena writes the numbers $1$, $2$, $3$, and $4$ in some order on a blackboard, such that she cannot swap two numbers and obtain the sequence $1$, $2$, $3$, $4$. How many sequences could she have written?
Let $S$ be the string $0101010101010$. Determine the number of substrings containing an odd number of $1$'s. (A substring is defined by a pair of (not necessarily distinct) characters of the string and represents the characters between, inclusively, the two elements of the string.)
The unit squares on the coordinate plane that have four lattice point vertices are colored black orwhite, as on a chessboard, shown on the diagram below. For an ordered pair $(m, n)$, let $OXZY$ be the rectangle with vertices $O = (0, 0)$, $X = (m, 0)$, $Z = (m, n)$ and $Y = (0, n)$. How many ordered pairs $(m, n)$ of nonzero integers exist such that rectangle $OXZY$ contains exactly 32 black squares?
In a $3 \times 4$ grid of 12 squares, find the number of paths from the top left corner to the bottom right corner that satisfy the following two properties:
- The path passes through each square exactly once.
- Consecutive squares share a side.
Two paths are considered distinct if and only if the order in which the twelve squares are visited is different. For instance, in the diagram below, the two paths drawn are considered the same.
A hand of four cards of the form $(c, c, c + 1, c + 1)$ is called a $tractor$. Vinjai has a deck consisting of four of each of the numbers $7$, $8$, $9$ and $10$. If Vinjai shuffles and draws four cards from his deck, compute the probability that they form a tractor.
How many ways are there to color the squares of a 10 by 10 grid with black and white such that in each row and each column there are exactly two black squares and between the two black squares in a given row or column there are exactly 4 white squares? Two configurations that are the same under rotations or reflections are considered different.
Link cuts trees in order to complete a quest. He must cut 3 Fenwick trees, 3 Splay trees and 3 KD trees. If he must also cut 3 trees of the same type in a row at some point during his quest, in how many ways can he cut the trees and complete the quest? (Trees of the same type are indistinguishable.)
Bessie shuffles a standard 52-card deck and draws five cards without replacement. She notices that all five of the cards she drew are red. If she draws one more card from the remaining cards in the deck, what is the probability that she draws another red card?