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Question
Three integers are chosen at random from the first 20 integers. The probability that their product is even is
Options
\[\frac{2}{19}\]
\[\frac{3}{29}\]
\[\frac{17}{19}\]
\[\frac{4}{19}\]
Solution
\[\frac{17}{19}\]
\[P\left( \text{ product is even } \right) = 1 - P\left( \text{ product is odd } \right)\]
\[\text{ For the product to be odd, the two digits must be odd } .\]
\[\text{ Now, 10 numbers are odd in the first 20 integers . } \]
\[ \therefore P\left( \text{ product is even} \right) = 1 - \frac{10}{20} \times \frac{9}{19} \times \frac{8}{18}\]
\[ = 1 - \frac{9}{38} \times \frac{8}{18}\]
\[ = 1 - \frac{2}{19}\]
\[ = \frac{17}{19}\]
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