In how many ways can the letters of the word UNIVERSAL be arranged when E,R,S are always together ?
शब्द UNIVERSAL के अक्षरों को कितने प्रकार से व्यवस्थित किया जा सकता है जब E,R,S हमेशा एक साथ हों?

Options

A. 31240
B. 32240
C. 30240
D. 29240

Answer

To solve the problem of arranging the letters of the word "UNIVERSAL" with the condition that the letters E, R, and S are always together, we can follow these steps:

### Step-by-Step Solution:

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

2. **Group the letters E, R, and S together:**
Since E, R, and S must always be together, we can treat them as a single entity or block. Let's denote this block as (ERS).

3. **Count the number of entities to arrange:**
After grouping E, R, and S, we now have the following entities to arrange:
- The block (ERS)
- The remaining letters: U, N, I, V, A, L

This gives us a total of 7 entities to arrange: (ERS), U, N, I, V, A, L.

4. **Calculate the arrangements of these entities:**
The number of ways to arrange 7 distinct entities is given by 7 factorial (7!):
\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \]

5. **Arrange the letters within the block (ERS):**
The letters E, R, and S can be arranged among themselves in 3 factorial (3!) ways:
\[ 3! = 3 \times 2 \times 1 = 6 \]

6. **Calculate the total arrangements:**
To find the total arrangements where E, R, and S are together, we multiply the arrangements of the entities by the arrangements within the block:
\[ \text{Total arrangements} = 7! \times 3! = 5040 \times 6 = 30240 \]

### Final Answer:
Thus, the total number of ways to arrange the letters of the word "UNIVERSAL" when E, R, and S are always together is **30,240**.

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