2010 AMC 10A Problems/Problem 18

Problem

Bernardo randomly picks 3 distinct numbers from the set $\{1,2,3,4,5,6,7,8,9\}$ and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set $\{1,2,3,4,5,6,7,8\}$ and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number?

$\textbf{(A)}\ \frac{47}{72} \qquad \textbf{(B)}\ \frac{37}{56} \qquad \textbf{(C)}\ \frac{2}{3} \qquad \textbf{(D)}\ \frac{49}{72} \qquad \textbf{(E)}\ \frac{39}{56}$

Solution

We can solve this by breaking the problem down into $2$ cases and adding up the probabilities.

Case $1$ : Bernardo picks $9$ . If Bernardo picks a $9$ then it is guaranteed that his number will be larger than Silvia's. The probability that he will pick a $9$ is $\dfrac{3}{9} = \frac{1}{3}$ .

Case $2$ : Bernardo does not pick $9$ . Since the chance of Bernardo picking $9$ is $\frac{1}{3}$ , the probability of not picking $9$ is $\frac{2}{3}$ .

If Bernardo does not pick 9, then he can pick any number from $1$ to $8$ . Since Bernardo is picking from the same set of numbers as Silvia, the probability that Bernardo's number is larger is equal to the probability that Silvia's number is larger.

Ignoring the $9$ for now, the probability that they will pick the same number is the number of ways to pick Bernardo's 3 numbers divided by the number of ways to pick any 3 numbers.

We get this probability to be $\frac{3!}{8\cdot{7}\cdot{6}} = \frac{1}{56}$

(This was confusing for me, so think of it like this: Bernardo chooses his number. Silva has one out of the total number of options to get the same number. The total number of options is 8 choose 3: $\frac{8\cdot{7}\cdot{6}}{3!} = \frac{1}{56}$ . There are 8 numbers and you choose three to make a three-digit number. Setting it up to be descending doesn't affect the number of ways there are. )

Probability of Bernardo's number being greater is $\frac{1-\frac{1}{56}}{2} = \frac{55}{112}$

Factoring the fact that Bernardo could've picked a $9$ but didn't:

$\frac{2}{3}\cdot{\frac{55}{112}} = \frac{55}{168}$

Adding up the two cases we get $\frac{1}{3}+\frac{55}{168} = \boxed{\frac{37}{56}\ \textbf{(B)}}$

2010 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 17	Followed by Problem 19
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2010 AMC 10A Problems/Problem 18

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