Life is A Test 2: Test Harder (Quite A Bit Harder)

by Brony_of_Brody

The Answer 33

The odds are about 0.017%.

For our purposes, a number can have digits in different combinations. All seven digits could be different. Or maybe they're all the same. Or maybe they have two digits that appear three times and a third only once. There are fifteen such combinations available.

For each type, we need to work out how many of the ten million numbers you have fall under this configuration, which gives us the probability Rainbow Dash has it. For example, there are 1,058,400 seven-digit phone numbers in which one digit is repeated three times, another appears twice, and two other digits appear once each. We could call that particular configuration 3A 2B C D — digit A appears three times, B twice, C and D once.

Then, we need to figure out how many exact scrambles there are. We can calculate this through the sum 7!/(3!2!1!1!)−1, for a total of 419 exact scrambles. (We minus 1 because we're not counting the ORIGINAL number.)

Finally, we can multiply the number of exact scrambles associated with each configuration by the number of your numbers with that configuration, summed up over the 15 possible configurations, and divided by 9.999999 x 10¹⁴ — that’s the number of possible numbers YOU have multiplied by the number of possible numbers RAINBOW has (one shy of 10,000,000 because one of your numbers is already taken).

It takes a while, but multiplying the possibilities for your numbers for each configuration (from 7A down to 1A 1B 1C 1D 1E 1F 1G) and multiplying them by the possible combinations for Rainbow's number, and then dividing by (10,000,000×9,999,999) gets us 0.0001676, or about 0.017%.