Using each of the digits 1 to 6, inclusive, exactly once, how many six-digit integers can be formed that are divisible by 6?
The game of Connex contains one 4-unit piece, two identical 3-unit pieces, three identical 2-unit pieces and four identical 1-unit pieces. How many different arrangements of pieces will make a 10-unit segment? The 10-unit segments consisting of the pieces 4-3-2-1 and 1-2-3-4 are two such arrangements to include.
How many positive two-digit integers have exactly $8$ positive factors?
A state license plate contains the state logo in the center, preceded by three letters and followed by three digits. If the first two letters must both be consonants, excluding Y, how many different license plates are possible?
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.
For how many two-element subsets {$a,b$} of the set {$1, 2, 3, \cdots, 36$} is the product of $ab$ a perfect square?
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?
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.)
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.
A code consists of four different digits from $1$ to $9$, inclusive. What is the probability of selection a code that consists of four consecutive digits but not necessarily in order? Express your answer as a common fraction.
In the diagram below, how many different routes are there from point $M$ to point $P$ using only the ling segments shown? A route is not allowed to intersect itself, not even at a single point.
Say that a rational number is special if its decimal expression is of the form $0.\overline{abcdef}$, where $a, b, c, d, e$ and $f$ are digits (possibly equal) that include each of the digits $2, 0, 1$, and $5$ at least once (in some order). How many special rational numbers are there?
Let $a$, $b$, $c$, $d$, and $e$ be five positive integers. If $ab+c=3115$, $c^2+d^2=e^2$, both $a$ and $c$ are prime numbers, $b$ is even and has $11$ divisors. Find these five numbers
How many three-digit numbers have at least one $2$ and at least one $3$?
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 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 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 $2019$?