Question:
A box contains 6 bottles of \(V_1\) drink, 3 bottles of \(V_2\) drink, and 4 bottles of \(V_3\) drink. If three bottles are drawn at random, find the probability that the three are not of the same variety.
A box contains 6 bottles of \(V_1\) drink, 3 bottles of \(V_2\) drink, and 4 bottles of \(V_3\) drink. If three bottles are drawn at random, find the probability that the three are not of the same variety.
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When finding the probability of "not all same," first calculate the probability of "all same" and subtract from 1. Use combinatorial counting for precise results.
Updated On: Jul 18, 2026
- \(\frac{6}{32}\)
- \(\frac{7}{13}\)
- \(\frac{8}{33}\)
- \(\frac{2}{7}\)
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The Correct Option is D
Solution and Explanation
Step 1: Total number of bottles.
Total bottles \(= 6 + 3 + 4 = 13\) bottles.
Step 2: Total ways to choose 3 bottles.
Total ways \(= \binom{13}{3} = \frac{13 \cdot 12 \cdot 11}{3 \cdot 2 \cdot 1} = 286\).
Step 3: Number of ways to choose 3 bottles of the same variety.
- For \(V_1\): \(\binom{6}{3} = 20\)
- For \(V_2\): \(\binom{3}{3} = 1\)
- For \(V_3\): \(\binom{4}{3} = 4\)
Total ways for same variety \(= 20 + 1 + 4 = 25\).
Step 4: Number of ways for 3 bottles not all the same.
Number of favorable outcomes \(= 286 - 25 = 261\).
Step 5: Probability calculation.
\[ P(\text{not all same}) = \frac{261}{286} = \frac{2}{7} \]
Step 6: Final conclusion.
Hence, the probability that the three bottles are not of the same variety is \[ \boxed{\frac{2}{7}} \]
Total bottles \(= 6 + 3 + 4 = 13\) bottles.
Step 2: Total ways to choose 3 bottles.
Total ways \(= \binom{13}{3} = \frac{13 \cdot 12 \cdot 11}{3 \cdot 2 \cdot 1} = 286\).
Step 3: Number of ways to choose 3 bottles of the same variety.
- For \(V_1\): \(\binom{6}{3} = 20\)
- For \(V_2\): \(\binom{3}{3} = 1\)
- For \(V_3\): \(\binom{4}{3} = 4\)
Total ways for same variety \(= 20 + 1 + 4 = 25\).
Step 4: Number of ways for 3 bottles not all the same.
Number of favorable outcomes \(= 286 - 25 = 261\).
Step 5: Probability calculation.
\[ P(\text{not all same}) = \frac{261}{286} = \frac{2}{7} \]
Step 6: Final conclusion.
Hence, the probability that the three bottles are not of the same variety is \[ \boxed{\frac{2}{7}} \]
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