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## Divisible?

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

 Puzzle ID: #42407 Fun: (2.28) Difficulty: (2.43) Category: Probability Submitted By: billy314

The digits one through nine are randomly arranged to make a number. What is the probability that the resulting number is divisible by eighteen?

If the digits one through nine are arranged, the resulting number will always be divisible by nine (a number is divisible by nine if the sum of its digits is divisible by nine). The question can be simplified to ask for the probability that the resulting number is even. Now, there are four options for the last digit (2, 4, 6, 8). There are then eight options for the digit before the last (one option is already taken). There are seven options for next, and so on. The total number of combinations divisible by eighteen is 4*8*7*6*5*4*3*2*1. The total number of combinations is 9*8*7*6*5*4*3*2*1. The probability a combination is divisible by eighteen is 4*8*7*6*5*4*3*2*1/9*8*7*6*5*4*3*2*1 or 4/9.

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