Question:

A bag contains 4 black, 6 yellow, and 8 red balls. Three balls are drawn at random from the bag. The probability that all of them are yellow is .......

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In probability problems, remember to divide the number of favorable outcomes by the total number of possible outcomes.
Updated On: Mar 10, 2025
  • \( \frac{7}{204} \)
  • \( \frac{11}{408} \)
  • \( \frac{3}{68} \)
  • \( \frac{5}{204} \)
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The Correct Option is D

Solution and Explanation

Step 1: Total number of balls
Total balls = 4 black + 6 yellow + 8 red = 18 balls.

Step 2: Total number of ways to select 3 balls from 18
The total number of ways to select 3 balls from 18 is given by \( \binom{18}{3} \).
\[ \binom{18}{3} = \frac{18 \times 17 \times 16}{3 \times 2 \times 1} = 816. \]

Step 3: Number of ways to select 3 yellow balls
Since there are 6 yellow balls, the number of ways to select 3 yellow balls from 6 is given by \( \binom{6}{3} \).
\[ \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20. \]

Step 4: Probability calculation
The probability that all 3 balls drawn are yellow is the ratio of the favorable outcomes to the total outcomes. \[ \text{Probability} = \frac{\binom{6}{3}}{\binom{18}{3}} = \frac{20}{816} = \frac{5}{204}. \]

Step 5: Final Answer
The correct answer is (d) \( \frac{5}{204} \). Final Answer: The correct answer is (d) \( \frac{5}{204} \).
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