2007 AMC 8 Problems/Problem 21

Revision as of 14:03, 3 November 2020 by Theepiccarrot7 (talk | contribs) (Solution 2)

Problem

Two cards are dealt from a deck of four red cards labeled $A$, $B$, $C$, $D$ and four green cards labeled $A$, $B$, $C$, $D$. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair?

$\textbf{(A)}\ \frac{2}{7}\qquad\textbf{(B)}\ \frac{3}{8}\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{4}{7}\qquad\textbf{(E)}\ \frac{5}{8}$

Video Solution

https://youtu.be/OOdK-nOzaII?t=1698

Solution 1

There are 4 ways of choosing a winning pair of the same letter, and $2 \left( \dbinom{4}{2} \right) = 12$ ways to choose a pair of the same color.

There's a total of $\dbinom{8}{2} = 28$ ways to choose a pair, so the probability is $\dfrac{4+12}{28} = \boxed{\textbf{(D)}\ \frac{4}{7}}$.

Solution 2

Notice that, no matter which card you choose, there are exactly 4 cards that either have the same color or letter as it. Since there are 7 cards left to choose from, the probability is $\frac{4}{7}$. theepiccarrot7

See Also

2007 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 20
Followed by
Problem 22
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AJHSME/AMC 8 Problems and Solutions

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