Practice (68)
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Sets $A$ and $B$, shown in the Venn diagram, have the same number of elements. Their union has $2007$ elements and their intersection has $1001$ elements. Find the number of elements in $A$.
Two cards are dealt from a deck of four red cards labeled $A$, $B$, $C$, $D$ and four green cards labeled $A$, $B$, $C$, $D$. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair?
A bag contains four pieces of paper, each labeled with one of the digits $1$, $2$, $3$ or $4$, with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. What is the probability that the three-digit number is a multiple of $3$?
Each of the $39$ students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twenty students have a dog and $26$ students have a cat. How many students have both a dog and a cat?
Three $\text{A's}$, three $\text{B's}$, and three $\text{C's}$ are placed in the nine spaces so that each row and column contain one of each letter. If $\text{A}$ is placed in the upper left corner, how many arrangements are possible?
Eight points are spaced around at intervals of one unit around a $2 \times 2$ square, as shown. Two of the $8$ points are chosen at random. What is the probability that the two points are one unit apart?
Ten tiles numbered $1$ through $10$ are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?
On a checkerboard composed of $64$ unit squares, what is the probability that a randomly chosen unit square does not touch the outer edge of the board?
The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime?
A three-digit integer contains one of each of the digits $1$, $3$, and $5$. What is the probability that the integer is divisible by $5$?
How many $3$-digit positive integers have digits whose product equals $24$?
How many non-congruent triangles have vertices at three of the eight points in the array shown below?
Square ABCD, shown here, has diagonals AC and BD that intersect at E. How many triangles of any size are in the figure?
The square in Figure 1 is cut along its diagonals creating four congruent triangles that then are arranged to create Figure 2. What is the probability that a randomly chosen point within the boundary of Figure 2 is in any of the shaded triangles? Express your answer as a common fraction.
If Desi flips a fair coin eight times, what is the probability that he will get the same number of heads and tails? Express your answer as a common fraction.
At Euclid Middle School the mathematics teachers are Miss Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, $8$ students in Mr. Newton's class, and $9$ students in Mrs. Young's class taking the AMC 8 this year. How many mathematics students at Euclid Middle School are taking the contest?
In a room, $2/5$ of the people are wearing gloves, and $3/4$ of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove?
Everyday at school, Jo climbs a flight of $6$ stairs. Joe can take the stairs $1$, $2$, or $3$ at a time. For example, Jo could climb $3$, then $1$, then $2$. In how many ways can Jo climb the stairs?
In a town of $351$ adults, every adult owns a car, motorcycle, or both. If $331$ adults own cars and $45$ adults own motorcycles, how many of the car owners do not own a motorcycle?
Bag $A$ has three chips labeled $1$, $3$, and $5$. Bag $B$ has three chips labeled $2$, $4$, and $6$. If one chip is drawn from each bag, how many different values are possible for the sum of the two numbers on the chips?
Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other?
A fair $6$-sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number?
How many $4$-digit positive integers have four different digits, where the leading digit is not zero, the integer is a multiple of $5$, and $5$ is the largest digit?
How many $4$-digit numbers greater than $1000$ are there that use the four digits of $2012$?
In the BIG N, a middle school football conference, each team plays every other team exactly once. If a total of $21$ conference games were played during the $2012$ season, how many teams were members of the BIG N conference?