Find the probability that in a random arrangement of letters of the words UNIVERSITY two I's do not come together.

Options

A. `(4)/(5)`
B. `(1)/(5)`
C. `(3)/(5)`
D. `(2)/(3)`

Answer

To solve the problem of finding 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: Count the total letters in the word "UNIVERSITY"
The word "UNIVERSITY" consists of the following letters:
- U
- N
- I
- V
- E
- R
- S
- I
- T
- Y

Counting these, we find there are a total of 10 letters.

**Hint:** Always start by counting the total number of items in the set.

### Step 2: Identify the total arrangements of the letters
The total number of arrangements of the letters in "UNIVERSITY" can be calculated using the formula for permutations of multiset:
\[ \text{Total arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \]
where \( n \) is the total number of letters, and \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeated letters.

In "UNIVERSITY," the letter I is repeated twice. Therefore:
\[ \text{Total arrangements} = \frac{10!}{2!} \]

**Hint:** Remember to account for repeated letters when calculating arrangements.

### Step 3: Calculate the total arrangements
Now, we compute:
\[ 10! = 3628800 \]
\[ 2! = 2 \]
Thus, the total arrangements are:
\[ \frac{3628800}{2} = 1814400 \]

**Hint:** Use a calculator for factorials if needed, especially for larger numbers.

### Step 4: Calculate arrangements where the two I's are together
To find the arrangements where the two I's are together, we can treat the two I's as a single unit. This means we now have the following units to arrange:
- II (the two I's together)
- U
- N
- V
- E
- R
- S
- T
- Y

This gives us a total of 9 units to arrange (II, U, N, V, E, R, S, T, Y).

The number of arrangements of these 9 units is:
\[ 9! = 362880 \]

**Hint:** When dealing with groups, consider them as single units to simplify the arrangement.

### Step 5: Calculate the probability that the two I's do not come together
The number of arrangements where the two I's do not come together is given by:
\[ \text{Arrangements where I's do not come together} = \text{Total arrangements} - \text{Arrangements where I's are together} \]
\[ = 1814400 - 362880 = 1451520 \]

Now, the probability that the two I's do not come together is:
\[ P(\text{I's not together}) = \frac{\text{Arrangements where I's do not come together}}{\text{Total arrangements}} \]
\[ = \frac{1451520}{1814400} \]

### Step 6: Simplify the probability
To simplify:
\[ P(\text{I's not together}) = \frac{1451520 \div 1814400} = \frac{81}{100} \]

**Hint:** Always simplify fractions to their lowest terms for clarity.

### Final Answer
The probability that in a random arrangement of the letters of the word "UNIVERSITY," the two I's do not come together is:
\[ \frac{81}{100} \]