The probability that in the random arrangement of the letters of the word `UNIVERSITY`, the two `I’`s does not come together is

Options

A. `4/5`
B. `1//5`
C. `1//10`
D. `9//10`

Answer

To find the probability that in a random arrangement of the letters of the word "UNIVERSITY", the two 'I's do not come together, we can follow these steps:

### Step 1: Calculate the Total Arrangements of the Letters
The word "UNIVERSITY" consists of 10 letters where the letter 'I' appears twice. The total arrangements (outcomes) can be calculated using the formula for permutations of multiset:

\[ \text{Total Arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots} \]

Where \(n\) is the total number of letters, and \(p_1, p_2, \ldots\) are the frequencies of the repeated letters.

For "UNIVERSITY":
- Total letters, \(n = 10\)
- The letter 'I' repeats 2 times.

Thus, the total arrangements are:

\[ \text{Total Arrangements} = \frac{10!}{2!} \]

### Step 2: Calculate the Arrangements Where the Two 'I's Come Together
To find the arrangements where the two 'I's are together, we can treat the two 'I's as a single unit or block. This reduces the problem to arranging the letters: {II, U, N, V, E, R, S, T, Y}, which gives us 9 units to arrange.

The total arrangements with 'II' together is:

\[ \text{Arrangements with II together} = 9! \]

### Step 3: Calculate the Favorable Outcomes
The favorable outcomes are the total arrangements minus the arrangements where the 'I's are together:

\[ \text{Favorable Outcomes} = \text{Total Arrangements} - \text{Arrangements with II together} \]

Substituting the values we calculated:

\[ \text{Favorable Outcomes} = \frac{10!}{2!} - 9! \]

### Step 4: Calculate the Probability
The probability that the two 'I's do not come together is given by:

\[ P(\text{I's not together}) = \frac{\text{Favorable Outcomes}}{\text{Total Arrangements}} \]

Substituting the values:

\[ P(\text{I's not together}) = \frac{\frac{10!}{2!} - 9!}{\frac{10!}{2!}} \]

### Step 5: Simplify the Expression
To simplify:

1. Calculate \(10!\) and \(9!\):
- \(10! = 10 \times 9!\)

2. Substitute \(10!\) in the probability expression:

\[ P(\text{I's not together}) = \frac{\frac{10 \times 9!}{2} - 9!}{\frac{10 \times 9!}{2}} \]

3. Factor out \(9!\):

\[ = \frac{9! \left(\frac{10}{2} - 1\right)}{\frac{10 \times 9!}{2}} = \frac{9! \left(5 - 1\right)}{\frac{10 \times 9!}{2}} = \frac{4}{5} \]

### Final Answer
Thus, the probability that the two 'I's do not come together is:

\[ \boxed{\frac{4}{5}} \]