How many three-digit numbers have at least one $2$ and at least one $3$?
A set of three points is randomly chosen from the grid shown. Each three point set has the same probability of being chosen. What is the probability that the points lie on the same straight line?
Cozy the Cat is going up a staircase of $10$ steps. She can either walk up $1$ step a time or jump $2$ steps a time. How many different ways can she reach the top of this staircase?
What is the area that is covered by putting a $8\times 6$ rectangle and a $5 \times 5$ square as shown on a table?
Restaurant MAS offers a set menu with $3$ choices of appetizers, $5$ choices of main dishes, and $2$ choices of desserts. How many possible combinations can a customer have for one appetizer, one main dish, and one dessert?
Eight chairs are arranged in two equal rows for a group of $8$. Joe and Mary must sit in the front row. Jack must sit in the back row. How many different seating plans can they have?
Two Britons, three Americans, and six Chinese form a line:
- How many different ways can the $11$ individuals line up?
- If two people of the same nationality cannot stand next to each other, how many different ways can the $11$ individuals line up?
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)
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?
How many different $6$-digit numbers can be formed by using digits $1$, $2$, and $3$, if no adjacent digits can be the same?
Joe wants to write $1$ to $n$ in a $1 \times n$ grid. The number 1 can be written in any grid, while the number $2$ must be written next to $1$ (can be at either side) so that these two numbers are together. The number 3 must be written next to this two-number block. This process goes on. Every new number written must stay next to the existing number block. How many different ways can Joe fill this $1 \times n$ grid?
How many positive divisors does $20$ have?
Find the number of different rectangles that satisfy the following conditions:
- Its area is $2015$
- The lengths of all its sides are integers
You have three colors {red; blue; green} with which you can color the faces of a regular octahedron ($8$ triangle sided polyhedron, which is two square based pyramids stuck together at their base), but you must do so in a way that avoids coloring adjacent pieces with the same color. How many different coloring schemes are possible? (Two coloring schemes are considered equivalent if one can be rotated to fit the other.)
How many different ways are there to cover a $1\times 10$ grid with some $1\times 1$ and $1\times 2$ pieces without overlapping?
What is the size of the largest subset $S'$ of $S=\{ 2^x 3^y 5^z : 0\le x,y,z \le 4\}$ such that there are no distinct elements $p,q \in S'$ with $p\mid q$.
Let $f(n)$ be the number of points of intersections of diagonals of a $n$-dimensional hypercube that is not the vertice of the cube. For example, $f(3) = 7$ because the intersection points of a cube's diagonals are at the centers of each face and the center of the cube. Find $f(5)$
Condier a $9\times 9$ grid of squares. Haraki fills each square in hthis grid with integer between 1 and 9, inclusive. The grid is called a $\textit{super-sudoku}$ if each of the following three conditions hold:
- Each column is this grid contains 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly once
- Each row is this grid contains 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly once
- Each $3\times 3$ sub-grid is this grid contains 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly once
How any such super-sudokus are there?
How many numbers between $1$ and $2020$ are multiples of $3$ or $4$ but not $5$?
How many positive integers, not exceeding $2019$, are relatively prime to $2019$?
Let $a$ and $b$ be two randomly selected points on a line segment of unit length. What is the probability that their distance is not more than $\frac{1}{2}$?
Randomly select $3$ real numbers $x$, $y$, and $z$ between 0 and 1. What is the probability that $x^2 + y^2 + z^2 > 1$?
There are several equally spaced parallel lines on a table. The distance between two adjacent lines is $2a$. On the table, toss a coin with a radius of $r$, $(r < a)$. Find the probability that the coin does not touch any line.
Joe breaks a $10$-meter long stick into three shorter sticks. Find the probability that these three sticks can form a triangle.
Break a stick into two parts. What is the probability that the length of one part is at least twice of that of the other?