This number's third power is the only cube in which three distinct digits each occur three times.
a. Jeopardy style: you provide the question.
b. What else can be said about this special number?
(In reply to What is... (solution)
Actually there are only 181440 nine-digit numbers have three occurrences of three distinct digits.
There are C(10,3) = 120 ways of choosing which 3 digits are used.
The number of ways of arranging the three digits in groups of 3 is 9!/(3!)^3 = 1680.
Allowing all arrangements, including those beginning with zero, gives 120*1680 = 201600.
We need to subtract those beginning with zero (otherwise we'd be choosing from a universe of 1,000,000,000 instead of 900,000,000):
There are C(9,2) = 36 choices of the other two numbers.
There are 8!/(((3!)^2)*2) = 560 ways of arranging the 8 digits past the initial zero for each of the 36 choices.
So we need to subtract 36*560 = 20160 from the initial 201600, leaving 181440, making the quoted expected number of cubes even smaller: .107856.
Posted by Charlie
on 2015-03-03 11:01:33