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Three numbers are chosen from the set {1, 2, 3, ..., 12}.  What is the probability that the product of the three numbers is divisible by 6?

 May 28, 2020
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1 - You make a 12 x 12 multiplication table like this:
(1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 14, 14, 15, 15, 16, 16, 16, 18, 18, 18, 18, 20, 20, 20, 20, 21, 21, 22, 22, 24, 24, 24, 24, 24, 24, 25, 27, 27, 28, 28, 30, 30, 30, 30, 32, 32, 33, 33, 35, 35, 36, 36, 36, 36, 36, 40, 40, 40, 40, 42, 42, 44, 44, 45, 45, 48, 48, 48, 48, 49, 50, 50, 54, 54, 55, 55, 56, 56, 60, 60, 60, 60, 63, 63, 64, 66, 66, 70, 70, 72, 72, 72, 72, 77, 77, 80, 80, 81, 84, 84, 88, 88, 90, 90, 96, 96, 99, 99, 100, 108, 108, 110, 110, 120, 120, 121, 132, 132, 144)

 

2 - Count the multiples of 6 like this:
(6, 6, 6, 6, 12, 12, 12, 12, 12, 12, 18, 18, 18, 18, 24, 24, 24, 24, 24, 24, 30, 30, 30, 30, 36, 36, 36, 36, 36, 42, 42, 48, 48, 48, 48, 54, 54, 60, 60, 60, 60, 66, 66, 72, 72, 72, 72, 84, 84, 90, 90, 96, 96, 108, 108, 120, 120, 132, 132, 144) >>Total = 60

 

3 - Probability = 60 / 144 = 5 / 12

 May 28, 2020

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