Henry Ernest Dudeney/Modern Puzzles/64 - Dividing by Eleven

Modern Puzzles by Henry Ernest Dudeney: $64$

Dividing by Eleven

If the $9$ digits are written at haphazard in any order,

for example $4 \ 1 \ 2 \ 5 \ 3 \ 9 \ 7 \ 6 \ 8$, what are the chances that the number that happens to be produced will be divisible by $11$ without remainder?

The number I have written at random is not, I see, so divisible, but if I had happened to make the $1$ and the $8$ change places it would be.

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Arithmetical and Algebraical Problems: Digital Puzzles: $64$. -- Dividing by Eleven
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Arithmetical and Algebraical Problems: Digital Puzzles: $115$. Dividing by Eleven