Question

A picture is divided into 6 squares of equal size. One of each of the squares is printed on each of the 6 faces of a cube. The cube is then copied until there are 6 identical cubes. If the 6 cubes are "rolled", find the probability that the upmost faces can be assembled into the original image.​

Answer 1

Explanation:

in other words : what is the probability when rolling 6 dice, that each die shows a different side ?

one shows a 1, one a 2, one a 3, ...

a probability is always the number of desired outcomes over the total number of possible outcomes.

the total number of results when rolling 6 dice is

6 possibilities for the first, 6 for the second, 6 for the third, ...

so,

6×6×6×6×6×6 = 6⁶ = 46,656

the desired cases (every die shows a different side) is represented by

6 possibilities for the first, then only 5 for the second (as one side is already taken by the first), 4 for the third, 3 for the fourth, 2 for the fifth, and 1 for the sixth.

so,

6×5×4×3×2×1 = 6! = 720

so, the probability to have all 6 distinct sides showing after rolling the 6 dice is

6! / 6⁶

or

6/6 × 5/6 × 4/6 × 3/6 × 2/6 × 1/6

that is

720/46,656 = 80/5184 = 5/324 = 0.015432099...