Question

In the word "UNIVERSITY", find the probability that the two "I"s do not come together.

• A. \( \frac{7}{11} \)
• B. \( \frac{8}{11} \)
• C. \( \frac{9}{11} \)
• D. \( \frac{10}{11} \)

Hint:

Remember: When letters repeat in a word, adjust the total number of arrangements by dividing by the factorial of the number of repeated letters. To calculate probabilities, consider the favorable and total outcomes.

Answer 1

The Correct Option is B

Step 1: Calculate total arrangements of "UNIVERSITY". The word "UNIVERSITY" has 10 letters. The letter "I" appears twice. The total number of distinct arrangements is computed as: \[ \text{Total arrangements} = \frac{10!}{2!} = \frac{3628800}{2} = 1814400 \] Step 2: Calculate arrangements with "I"s together. Treat the two "I"s as a single unit. We now arrange 9 units: "II", U, N, V, E, R, S, T, Y. The number of arrangements for these 9 units is: \[ \text{Arrangements with "I"s together} = 9! = 362880 \] Step 3: Calculate arrangements with "I"s not together. Subtract arrangements where "I"s are together from total arrangements: \[ \text{Arrangements with "I"s not together} = 1814400 - 362880 = 1451520 \] Step 4: Calculate the probability of "I"s not being together. The probability is the ratio of arrangements where "I"s are not together to the total arrangements: \[ \text{Probability} = \frac{1451520}{1814400} = \frac{8}{11} \] Answer: The probability that the two "I"s do not occur together is \( \frac{8}{11} \).