Statistics And Probability Question 217
Question: There are n different objects 1, 2, 3,……n distributed at random in n places marked 1, 2, 3, ……n. The probability that at least three of the objects occupy places corresponding to their number is
Options:
A) $ \frac{1}{6} $
B) $ \frac{5}{6} $
C) $ \frac{1}{3} $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
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Let $ E_{i} $ denote the event that the $ i^{th} $ object goes to the $ i^{th} $ place, we have $ P(E_{i})=\frac{(n-1)\,!}{n\,!}=\frac{1}{n},\forall i $ and $ P(E_1\cap E_{j}\cap E_{l})=\frac{(n-3)!}{n!} $ for $ i<j<k $ Since we can choose 3 places out of n in $ {}^{n}C_3 $ ways. The probability of the required event is $ {}^{n}C_3.\frac{(n-3)\,!}{n\,!}=\frac{1}{6} $ .
BETA
