Practice (71)

2477

Use digits $1$, $2$, $3$, $4$, and $5$ without repeating to create a number.

How many 5-digit numbers can be formed?
How many numbers will have the two even digits appearing between $1$ and $5$? (e.g.12345)

2478

Joe plans to put a red stone, a blue stone, and a black stone on a $10 \times 10$ grid. The red stone and the blue stone cannot be in the same column. The blue stone and the black stone cannot be in the same row. How many different ways can Joe arrange these three stones?

2481

How many positive divisors does $20$ have?

2567

How many integers between $1000$ and $9999$ have four distinct digits?

2569

How many pairs of parallel edges, such as $\overline{AB}$ and $\overline{GH}$ or $\overline{EH}$ and $\overline{FG}$, does a cube have?

2718

How many positive integers greater than $9$ are there such that every digit is less that the digit on its right?

2723

How many fraction numbers between $0$ and $1$ are there whose denominator is $1001$ when written in its simplest form?

2729

After having taken the same exam, Joe found he answered 1/3 of total problems incorrectly. Mary answered 6 incorrectly. The problems both didn't get right accounts for 1/5 of the total. Can you find how many problems did they both get right?

2881

Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\tfrac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N) < \tfrac{321}{400}$?

2967

A regular icosahedron is a $20$-faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top vertex and all in the same horizontal plane, and a lower pentagon of five vertices all adjacent to the bottom vertex and all in another horizontal plane. Find the number of paths from the top vertex to the bottom vertex such that each part of a path goes downward or horizontally along an edge of the icosahedron, and no vertex is repeated.

3046

How many rectangles of any size are in the grid shown here?

3369

The "Middle School Eight" basketball conference has $8$ teams. Every season, each team plays every other conference team twice (home and away), and each team also plays $4$ games against non-conference opponents. What is the total number of games in a season involving the "Middle School Eight" teams?

3695

Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions?

3764

The number $21!=51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd?

3828

Find the number of possible arrangements in Fisher Random Chess. The diagram below is one possible arrangement.

In a legal arrangement, the White's position must satisfy the following criteria:

Eight pawns must be in the $2^{nd}$ row. (The same as regular chess)
Two bishops must be in opposite colored squares (e.g. $b1$ and $e1$ in the above diagram)
King must locate between two rooks (e.g. in the diagram above, King is at $c1$ and two rooks are at $a1$ and $g1$)

The Black's position will be mirroring to the White's.

4729

How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$

4746

How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n$?

4770

Randomly draw a card twice with replacement from $1$ to $10$, inclusive. What is the probability that the product of these two cards is a multiple of $7$?

4771

How many even $4$- digit integers are there whose digits are distinct?

AdditionPrinciple MultiplicationPrinciple CountTheOpposite BasicProbability NumberTheoryBasic

AMC8 AMC10/12 AIME

With Solutions

back to index | new