examday

/TMUA

A white wooden cube is made of $n^3$ identical, smaller cubes, where $n$ is an even number. Each small cube on the surface of the large cube is numbered with a “0”. The numbered cubes are then removed, and the small cubes on the new surface of the remaining cube are all numbered with “1”. The process repeats, so that we have small cubes with numbers $0,1,2,3,\ldots$

The wooden block is now dismantled into the $n^3$ constituent cubes and put into a bag. Small cubes are picked consecutively at random without replacement. As $n$ increases, which of the following is true of the probability that all the numbers from $0$ to $\dfrac n2$ inclusive have been picked before any number repeats itself?