Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3, \dots, 2010\}$. What is the probability that $abc ab a$ is divisible by 3
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The probability that abc ab a is divisible by 3 is 13/27.We group this into groups of 3, because 3|2010. This means that every residue class mod 3 has an equal probability.If 3|a, we are done. There is a probability of 1/3 that that happens.Otherwise, we have 3 | bc + b + 1, which means that b (c + 1 ) = 2 mod{3}. So eitherb = 1 mod (3) c = 1 mod (3)orb = 2 mod (3) c = 0 mod (3)Which will lead to the property being true. There is a 1/3 * 1/3 = 1/9 chance for each bundle of cases to be true.Thus, the total for the cases is 2/9. But we have to multiply by 2/3 because this only happens with a 2/3 chance. So the total is actually 4/27.The grand total is1/3 + 4/27 = 13/27Learn more about probability here: https://brainly.com/question/24756209#SPJ4