There are only five possible shapes for a die, if the die is to be considered fair. A fair die means each face has the same shape and the same possibility of appearing on every roll. The five possible shapes are
Tetrahedron: 4 faces, each face is an equilateral triangle
Cube: 6 faces, each face is a square
Octahedron: 8 faces, each face is an equilateral triangle
Dodecahedron: 12 faces, each face is a regular pentagon
Icosahedron: 20 faces, each face is an equilateral triangle
If you had five dice, each die being one of the five shapes above with each side containing a number, how many combinations are possible if you were to roll the five dice at the same time?

Problems and applications q5 four roommates are planning to spend the weekend in their dorm room watching old movies, and they are debating how many to watch. here is their willingness to pay for each film: willingness to pay (dollars) shen antonio dmitri jake first film 10 9 8 3 second film 8 7 6 2 third film 6 5 4 1 fourth film 4 3 2 0 fifth film 2 1 0 0 within the dorm room, the showing of a movie a public good. if it costs $12 to rent a movie, the roommates should rent movies in order to maximize the total surplus. suppose the roommates choose to rent the optimal number of movies you just indicated and then split the cost of renting equally. this means that each roommate will pay $ .