MathDB

21

Part of 2011 AMC 10

Problems(2)

Probability of selecting 4 genuine coins

Source: 2011 AMC A Problem 21

6/25/2011
Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the 10 coins. A second pair is selected at random without replacement from the remaining 8 coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all 4 selected coins are genuine?
<spanclass=′latex−bold′>(A)</span> 711<spanclass=′latex−bold′>(B)</span> 913<spanclass=′latex−bold′>(C)</span> 1115<spanclass=′latex−bold′>(D)</span> 1519<spanclass=′latex−bold′>(E)</span> 1516 <span class='latex-bold'>(A)</span>\ \frac{7}{11}\qquad<span class='latex-bold'>(B)</span>\ \frac{9}{13}\qquad<span class='latex-bold'>(C)</span>\ \frac{11}{15}\qquad<span class='latex-bold'>(D)</span>\ \frac{15}{19}\qquad<span class='latex-bold'>(E)</span>\ \frac{15}{16}
probabilityconditional probabilityAMC
Pairwise differences of numbers

Source: AMC 10 2011 b Problem 21

2/24/2011
Brian writes down four integers w>x>y>zw > x > y > z whose sum is 4444. The pairwise positive differences of these numbers are 1,3,4,5,6,1,3,4,5,6, and 99. What is the sum of the possible values for ww?
<spanclass=′latex−bold′>(A)</span> 16<spanclass=′latex−bold′>(B)</span> 31<spanclass=′latex−bold′>(C)</span> 48<spanclass=′latex−bold′>(D)</span> 62<spanclass=′latex−bold′>(E)</span> 93 <span class='latex-bold'>(A)</span>\ 16 \qquad <span class='latex-bold'>(B)</span>\ 31 \qquad <span class='latex-bold'>(C)</span>\ 48 \qquad <span class='latex-bold'>(D)</span>\ 62 \qquad <span class='latex-bold'>(E)</span>\ 93
AMC