MathDB

18

Part of 2010 AMC 10

Problems(2)

Bernardo and Silvia counts from sets

Source: AMC 12A 2010, Problem 16

2/10/2010
Bernardo randomly picks 3 3 distinct numbers from the set {1,2,3,4,5,6,7,8,9} \{1,2,3,4,5,6,7,8,9\} and arranges them in descending order to form a 3 3-digit number. Silvia randomly picks 3 3 distinct numbers from the set {1,2,3,4,5,6,7,8} \{1,2,3,4,5,6,7,8\} and also arranges them in descending order to form a 3 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number? <spanclass=′latex−bold′>(A)</span> 4772<spanclass=′latex−bold′>(B)</span> 3756<spanclass=′latex−bold′>(C)</span> 23<spanclass=′latex−bold′>(D)</span> 4972<spanclass=′latex−bold′>(E)</span> 3956 <span class='latex-bold'>(A)</span>\ \frac {47}{72}\qquad <span class='latex-bold'>(B)</span>\ \frac {37}{56}\qquad <span class='latex-bold'>(C)</span>\ \frac {2}{3}\qquad <span class='latex-bold'>(D)</span>\ \frac {49}{72}\qquad <span class='latex-bold'>(E)</span>\ \frac {39}{56}
probabilityAMC
Probability of being divisible by 3

Source: AMC 12B 2010, Problem 16

2/25/2010
Positive integers a,b, a,b, and c c are randomly and independently selected with replacement from the set {1,2,3,…,2010}. \{ 1,2,3,\dots,2010 \}. What is the probability that abc \plus{} ab \plus{} a is divisible by 3 3? <spanclass=′latex−bold′>(A)</span> 13<spanclass=′latex−bold′>(B)</span> 2981<spanclass=′latex−bold′>(C)</span> 3181<spanclass=′latex−bold′>(D)</span> 1127<spanclass=′latex−bold′>(E)</span> 1327 <span class='latex-bold'>(A)</span>\ \dfrac{1}{3} \qquad<span class='latex-bold'>(B)</span>\ \dfrac{29}{81} \qquad<span class='latex-bold'>(C)</span>\ \dfrac{31}{81} \qquad<span class='latex-bold'>(D)</span>\ \dfrac{11}{27} \qquad<span class='latex-bold'>(E)</span>\ \dfrac{13}{27}
probabilityAMC