The number of ways in which the letters of the word 'ARRANGE' can be arranged so that two A's are together is

Options

A. 160
B. 200
C. 360
D. 900

Answer

To find the number of ways in which the letters of the word "ARRANGE" can be arranged such that the two A's are together, we can follow these steps:

### Step-by-Step Solution:

1. **Treat the two A's as a single unit**: Since we want the two A's to be together, we can treat them as one single entity or block. So, instead of "A A R R A N G E", we can think of it as "AA R R N G E".

2. **Count the total units**: After treating the two A's as one unit, we now have the following units to arrange: "AA", "R", "R", "N", "G", "E". This gives us a total of 6 units.

3. **Identify repeating units**: In these 6 units, the letter R appears twice. Thus, we need to account for the repetitions when calculating the arrangements.

4. **Calculate the arrangements**: The formula for the arrangements of n items where some items are repeated is given by:
\[ \text{Number of arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots} \]
where \( n \) is the total number of items, and \( p_1, p_2, \ldots \) are the frequencies of the repeated items.

Here, we have:
- Total units (n) = 6 (AA, R, R, N, G, E)
- Repeated units (R) = 2

Therefore, the number of arrangements is:
\[ \text{Number of arrangements} = \frac{6!}{2!} \]

5. **Calculate the factorials**:
- \( 6! = 720 \)
- \( 2! = 2 \)

Now, substituting these values:
\[ \text{Number of arrangements} = \frac{720}{2} = 360 \]

Thus, the number of ways in which the letters of the word "ARRANGE" can be arranged such that the two A's are together is **360**.