Cause and Effect Mr.11 & divisibility!

We call a natural number \(n\) as 'awesome' if at least one of the numbers formed by permutations of digits of \(n\) is divisible by 11.

There are \(x\) 'awesome' numbers and \(y\) 'non-awesome' numbers less than 1001.

Find the ratio \(\dfrac{y}{x}\) (as a decimal value).

Details and assumptions:

  • All the numbers are represented in base 10.

  • If a number already has zeroes, they can be brought to the start during permutations. e.g \(1102\) can be permuted to \(0121\)

  • You can't add zeroes at the start of the number like \(01201\) being permuted to \(12100\) is not allowed, because the original number is intended to be \(1201\) and not \(01201\).

  • A number is 'non-awesome' if it is not awesome, i.e. none of its permutations is divisible by 11.

This is a part of the set 11≡ awesome (mod remainders)

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