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 to arrange the letters of the word "ARRANGE" such that the two A's are together, we can follow these steps:

### Step 1: Treat the two A's as a single unit
Since we want the two A's to be together, we can consider them as a single entity or unit. Therefore, we can represent the two A's as "AA".

### Step 2: Count the total units
Now, instead of the original letters A, R, R, A, N, G, E, we have the following units to arrange:
- AA (the combined A's)
- R
- R
- N
- G
- E

This gives us a total of 6 units to arrange: AA, R, R, N, G, E.

### Step 3: Calculate the arrangements
To find the number of arrangements of these 6 units, we use the formula for permutations of multiset:
\[ \text{Number of arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots} \]
where \( n \) is the total number of units and \( p_1, p_2, \ldots \) are the frequencies of the repeating units.

In our case:
- Total units \( n = 6 \)
- The letter R repeats 2 times.

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

### Step 4: Calculate factorials
Now, we calculate \( 6! \) and \( 2! \):
- \( 6! = 720 \)
- \( 2! = 2 \)

### Step 5: Final calculation
Now we can substitute these values into our formula:
\[ \text{Number of arrangements} = \frac{720}{2} = 360 \]

### Conclusion
Therefore, 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**.

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