Question

A 12 by 12 by 12 cube has all its faces painted blue. It is then broken down to 1728 individual 1 by 1 by 1 cubes. The probability that a randomly chosen 1 by 1 by 1 cube has exactly 2 faces painted blue is \frac {a} {b} , where a, b are coprime. What is the value of a+ b?

Answer 1

There will be 12*(12 - 2) = 120 cubes with exactly 2 faces painted blue.

ie 120/1728 = 5/72
ie a + b = 5 + 72 =77