Number of ways in which the letters of the word ‘B U L B U L’ can be arranged in a line in any order is also equal to the:

Options

A. (a) Number of ways in which 2 alike Apples and 4 alike Mangoes can be distributed in 3 children so that each child receives any number of fruits.
B. (b) Number of ways in which 6 different books can be tied up into 3 bundles, if each bundle is to have equal number of books.
C. (c) Coefficient of `x^(2)y^(2)z^(2)` in the expansion of `(x+y+z)^(6)`
D. (d) Number of ways in which 6 different prizes can be distributed equally in three children.

Answer

To find the number of ways in which the letters of the word "BULBUL" can be arranged, we will follow these steps:

### Step 1: Identify the letters and their frequencies
The word "BULBUL" consists of the letters:
- B: 2
- U: 2
- L: 2

### Step 2: Calculate the total number of letters
The total number of letters in the word "BULBUL" is 6 (B, U, L, B, U, L).

### Step 3: Use the formula for permutations of multiset
The formula for the number of arrangements of letters when there are repeated letters is given by:

\[ \text{Number of arrangements} = \frac{n!}{n_1! \times n_2! \times n_3!} \]

where:
- \( n \) is the total number of letters,
- \( n_1, n_2, n_3 \) are the frequencies of the repeated letters.

In our case:
- \( n = 6 \) (total letters),
- \( n_1 = 2 \) (for B),
- \( n_2 = 2 \) (for U),
- \( n_3 = 2 \) (for L).

### Step 4: Substitute the values into the formula
Now we substitute the values into the formula:

\[ \text{Number of arrangements} = \frac{6!}{2! \times 2! \times 2!} \]

### Step 5: Calculate the factorials
Now, we calculate the factorials:
- \( 6! = 720 \)
- \( 2! = 2 \)

### Step 6: Perform the calculations
Now we substitute these values into our equation:

\[ \text{Number of arrangements} = \frac{720}{2 \times 2 \times 2} = \frac{720}{8} = 90 \]

### Final Answer
The number of ways in which the letters of the word "BULBUL" can be arranged is **90**.

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