How many ways can the letters of the word 'UNIVERSAL' be arranged? In how many of these of E, R, S always occur together?

Options

A. 32240
B. 30240
C. 30248
D. 31240

Answer

To solve the problem of arranging the letters of the word "UNIVERSAL" and finding how many arrangements have the letters E, R, S always occurring together, we can break it down into steps:

### Step 1: Count the total letters in "UNIVERSAL"
The word "UNIVERSAL" consists of 9 letters: U, N, I, V, E, R, S, A, L.

### Step 2: Identify any repeating letters
In the word "UNIVERSAL", there are no repeating letters. Each letter is unique.

### Step 3: Calculate the total arrangements of the letters
Since all letters are unique, the total number of arrangements of the letters in "UNIVERSAL" can be calculated using the factorial of the number of letters:

\[ \text{Total arrangements} = 9! = 362880 \]

### Step 4: Consider E, R, S as a single unit
To find the arrangements where E, R, and S are always together, we can treat E, R, and S as a single unit or block. This block can be represented as (ERS).

### Step 5: Count the total units
Now, instead of 9 letters, we have the following units:
- (ERS)
- U
- N
- I
- V
- A
- L

This gives us a total of 7 units (the block ERS counts as one unit).

### Step 6: Calculate arrangements of these units
The total arrangements of these 7 units can be calculated as:

\[ \text{Arrangements of units} = 7! = 5040 \]

### Step 7: Arrange E, R, S within their block
Within the block (ERS), the letters E, R, and S can be arranged among themselves. The number of arrangements of E, R, and S is:

\[ \text{Arrangements of (ERS)} = 3! = 6 \]

### Step 8: Calculate total arrangements with E, R, S together
Now, we multiply the arrangements of the units by the arrangements within the block:

\[ \text{Total arrangements with E, R, S together} = 7! \times 3! = 5040 \times 6 = 30240 \]

### Final Answer
1. The total arrangements of the letters in "UNIVERSAL" is **362880**.
2. The number of arrangements where E, R, S are always together is **30240**.