If there are only a handful of objects, then you can count them with a moment's thought, but the techniques of combinatorics can extend to quickly and efficiently tabulating astronomical quantities.
Adam, Billy, Cathy, and David are in a race. We know that
At the end, their names are written on the scoreboard in the order in which they finished the race, from first place to last place. In how many different ways could their names be arranged?
Amy, Billy, and Catherine are standing in line and Amy is best friends with Catherine so they want to stand together, but Amy hates Billy so they can’t stand together. Is it possible for them to all be happy with an ordering?
A tetromino is defined as a plane geometric figure made by joining four equal squares edge to edge. (Some samples are shown above; not every possibility is pictured.) If we say two tetrominos are the same if they can overlap perfectly after some series of rotations and/or flips, how many distinct tetrominos are there?
You have three colors of paint, and you want to paint a regular tetrahedron (a polyhedra with 4 faces that are all equilateral triangles). How many ways are there to do so if you only paint each face one of your three colors and two colored tetrahedrons are considered equivalent if they can be rotated to look the same?
Consider a simplified Sudoku played on a three by three grid. The numbers 1 through 3 must be placed into the grid such that in every column and every row there is one of each number. Given that there are no numbers initially placed into the grid, how many ways are there to fill out the puzzle?