Difference between revisions of "2016 AMC 10B Problems/Problem 12"

Latest revision as of 19:16, 31 August 2025

Problem

Two different numbers are selected at random from $\{1, 2, 3, 4, 5\}$ and multiplied together. What is the probability that the product is even?

$\textbf{(A)}\ 0.2\qquad\textbf{(B)}\ 0.4\qquad\textbf{(C)}\ 0.5\qquad\textbf{(D)}\ 0.7\qquad\textbf{(E)}\ 0.8$

Solution 1

The product will be even if at least one selected number is even, and odd if none are. Using complementary counting, the chance that both numbers are odd is $\frac{\tbinom32}{\tbinom52}=\frac3{10}$ , so the answer is $1-0.3$ which is $\boxed{\textbf{(D) }0.7}$ .

An alternate way to finish: Since it is odd if none are even, the probability is $1-(\frac{3}{5} \cdot \frac{2}{4})=1-\frac{3}{10}=0.7 \Longrightarrow \boxed{\textbf{(D) }0.7}$ . ~Alternate solve by JH. L

Solution 2

There are $2$ cases to get an even number. Case 1: $\text{Even} \times \text{Even}$ and Case 2: $\text{Odd} \times \text{Even}$ . Thus, to get an $\text{Even} \times \text{Even}$ , you get $\frac {\binom {2}{2}}{\binom {5}{2}}= \frac {1}{10}$ . And to get $\text{Odd} \times \text{Even}$ , you get $\frac {\binom {3}{1} \binom {2}{1}}{\binom {5}{2}}= \frac {6}{10}$ . $\frac {1}{10}+\frac {6}{10}=\frac {7}{10}$ which is $0.7$ and the answer is $\boxed{\textbf{(D) }0.7}$ .

Solution 3

Note that we have three cases to get an even number: even $\times$ even, odd $\times$ even and even $\times$ odd. The probability of case 1 is $\dfrac{2}{5}\cdot\dfrac{1}{4}$ , the probability of case 2 is $\dfrac{2}{5}\cdot\dfrac{3}{4}$ and the probability of case 3 is $\dfrac{3}{5}\cdot\dfrac12$ .

Adding these up we get $\dfrac{1}{10}+\dfrac{3}{10}+\dfrac{3}{10} = \boxed{\textbf{(D) }0.7}.$

-ConfidentKoala4

Video Solution by OmegaLearn

https://youtu.be/IRyWOZQMTV8?t=933

~ pi_is_3.14

Video Solution

https://youtu.be/tUpKpGmOwDQ - savannahsolver

@@ Line 13: / Line 13: @@
 ==Solution 2==
-There are <math>2</math> cases to get an even number. Case 1: <math>\text{Even} \times \text{Even}</math> and Case 2: <math>\text{Odd} \times \text{Even}</math>. Thus, to get an <math>\text{Even} \times \text{Even}</math>, you get <math>\frac {\binom {2}{2}}{\binom {5}{2}}= \frac {1}{10}</math>. And to get <math>\text{Odd} \times \text{Even}</math>, you get <math>\frac {\binom {3}{1}}{\binom {5}{2}}= \frac {6}{10}</math>. <math>\frac {1}{10}+\frac {6}{10}=\frac {7}{10}</math> which is <math>0.7</math> and the answer is <math>\boxed{\textbf{(D) }0.7}</math>.
+There are <math>2</math> cases to get an even number. Case 1: <math>\text{Even} \times \text{Even}</math> and Case 2: <math>\text{Odd} \times \text{Even}</math>. Thus, to get an <math>\text{Even} \times \text{Even}</math>, you get <math>\frac {\binom {2}{2}}{\binom {5}{2}}= \frac {1}{10}</math>. And to get <math>\text{Odd} \times \text{Even}</math>, you get <math>\frac {\binom {3}{1} \binom {2}{1}}{\binom {5}{2}}= \frac {6}{10}</math>. <math>\frac {1}{10}+\frac {6}{10}=\frac {7}{10}</math> which is <math>0.7</math> and the answer is <math>\boxed{\textbf{(D) }0.7}</math>.
 ==Solution 3==

2016 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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