Recent questions tagged combinatory

The squares of a chessboard are painted 8 different colors. The squares of each row are painted all 8 colors and no 2 consecutive squares in one column can be painted the...

Find the number of derangements of the integers from 1 to 10 inclusive, satisfying the condition that the set of elements in the 1st five places is:(1) 1,2,3,4,5 in some ...

A supplier of art material has four reams of handmade paper, three boxes of acrylic colors and two printing blocks. The two artists in the shop want to buy one item each,...

There are $5$ friends $A, B, C, D, E$. Because of a heated argument not all of them are on speaking terms anymore. The array below describes who is talking to whom. A val...

A domino is a sheet of paper with one number from $\{0,1,2,3,4,5,6,7,8,9\}$ printed on each half. For example, a $4-4$ domino has $4$ printed on both halves. A $0-4$ domi...

Consider a city with $n$ East-West Streets (EWS) and $n$ North-South Avenues (NSA). The EWS are the line segments $\{y=j \mid 1 \leq x \leq n\}$ for all $j \in\{1,2, \ldo...

Towns $A, B$ and $C$ are connected to each other by several roads, with at least one road connecting each pair. In going from one town to another, one can take the road t...

Solitaire Tic-Tac-Toe is a new game on the market. Instead of adding X's and O's to an empty $3 \times 3$ grid, you start with a $3 \times 3$ grid in which every position...

A perfect shuffle of a deck of cards divides the deck into two equal parts and then interleaves the cards from each half, starting with the first card of the first half.F...

In a football league, a collection of teams play against each other a number of times. A team gets $3$ points for each match that it wins, $1$ point for each match that i...

A new game show on TV has $100$ boxes numbered $1,2, \ldots, 100$, each containing a mystery prize. The prizes are of different types, $a, b, c, \ldots$, in decreasing or...

Let $a_{1}, \ldots, a_{n}$ be integers. Show that for some $k, m$ such that $1 \leq k \leq m \leq n$, the sum $a_{k}+a_{k+1}+\cdots+a_{m}$ is divisible by $n$. (Hint: Con...

How many elements are in the following set?\[\{(A, B) \mid A \subseteq B \subseteq\{1,2,3, \ldots, n\}\}\]$2^{n-1}$$3^{n}$$2^{n+1}$$2^{2 n}$