We call a \(10\)-digit number
interesting if it is divisible by \(11111\), and all its digits are different.
How many interesting numbers does there exist?
Note that a number \(k = a_0 + 10a_1 + \dots
+10^9 a_9\) is divisible by \(11111\) if and only if a number \(m = (a_0+a_5) +10(a_1+a_6) + \dots + 10^4
(a_4+a_9)\) is also divisible by \(11111\). This is because \(100000=1+9 \times 11111\) and we subtract
\(99999 (a_5 + 10a_6 + 100a_7 + 1000a_8
+10000a_9)\) from the original number.