Question:
The probability that in a random arrangement of the letters of the word 'UNIVERSITY', the two 'I's do not come together is
The probability that in a random arrangement of the letters of the word 'UNIVERSITY', the two 'I's do not come together is
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Find the probability that the two I's come together and subtract it from 1.
Updated On: Nov 13, 2026
- \(\frac{1}{5}\)
- \(\frac{1}{10}\)
- \(\frac{4}{5}\)
- \(\frac{3}{10}\)
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept
UNIVERSITY has 10 letters with the letter I repeated twice.
Step 2: Total Arrangements
Total arrangements \(=\dfrac{10!}{2!}\).
Step 3: I's Together
Treat II as one block: 9 objects, so \(9!\) arrangements.
\(P(\text{together})=\dfrac{9!}{10!/2}=\dfrac{2}{10}=\dfrac{1}{5}\).
Step 4: Result
\(P(\text{not together})=1-\dfrac{1}{5}=\dfrac{4}{5}\), option (C).
Final Answer:
The probability that the two I's are not together is \(\dfrac{4}{5}\). \[ \boxed{\text{(C) }\dfrac{4}{5}} \]
UNIVERSITY has 10 letters with the letter I repeated twice.
Step 2: Total Arrangements
Total arrangements \(=\dfrac{10!}{2!}\).
Step 3: I's Together
Treat II as one block: 9 objects, so \(9!\) arrangements.
\(P(\text{together})=\dfrac{9!}{10!/2}=\dfrac{2}{10}=\dfrac{1}{5}\).
Step 4: Result
\(P(\text{not together})=1-\dfrac{1}{5}=\dfrac{4}{5}\), option (C).
Final Answer:
The probability that the two I's are not together is \(\dfrac{4}{5}\). \[ \boxed{\text{(C) }\dfrac{4}{5}} \]
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