Henry Ernest Dudeney/Modern Puzzles/66 - Another 37 Division

Modern Puzzles by Henry Ernest Dudeney: $66$

Another $37$ Division

Here is an interesting extension of the last puzzle.

If the $9$ digits are written at haphazard in any order, for example $412539768$,

what are the chances that the number that happens to be produced will be divisible by $37$ without remainder?

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Arithmetical and Algebraical Problems: Digital Puzzles: $66$. -- Another $37$ Division
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Arithmetical and Algebraical Problems: Digital Puzzles: $117$. Another $37$ Division