12 boys and 2 girls are to be seated in a row such that there are atleast 3 boys between the 2 girls. The number of ways this can be done is `lamdaxx12!`. The value of `lamda` is

Options

A. 55
B. 110
C. 20
D. 45

Explanation

To solve the problem of seating 12 boys and 2 girls in a row such that there are at least 3 boys between the two girls, we can follow these steps:

### Step 1: Total Arrangement without Restrictions
First, we calculate the total number of arrangements of 12 boys and 2 girls without any restrictions. The total number of people is 14 (12 boys + 2 girls). The total arrangements can be calculated using the factorial:

\[ \text{Total arrangements} = 14! \]

### Step 2: Arrangement with Girls Together
Next, we consider the scenario where the two girls are treated as a single unit or block. This means we have 12 boys and 1 block of girls, making a total of 13 units to arrange. The arrangements can be calculated as:

\[ \text{Arrangements with girls together} = 13! \]

Since the two girls can be arranged among themselves in 2 ways, we multiply by 2:

\[ \text{Total arrangements with girls together} = 2 \times 13! \]

### Step 3: Arrangement with 1 Boy Between Girls
Now, we consider the case where there is exactly 1 boy sitting between the two girls. In this case, we treat the girls and the boy between them as a block. This gives us 11 boys and 1 block (2 girls + 1 boy), resulting in 12 units to arrange. The arrangements can be calculated as:

\[ \text{Arrangements with 1 boy between girls} = 12! \]

Since the two girls can be arranged among themselves in 2 ways, we multiply by 2:

\[ \text{Total arrangements with 1 boy between girls} = 2 \times 12! \]

### Step 4: Arrangement with 2 Boys Between Girls
Next, we consider the case where there are exactly 2 boys sitting between the two girls. In this case, we treat the girls and the two boys between them as a block. This gives us 10 boys and 1 block (2 girls + 2 boys), resulting in 11 units to arrange. The arrangements can be calculated as:

\[ \text{Arrangements with 2 boys between girls} = 11! \]

Again, the two girls can be arranged among themselves in 2 ways, so we multiply by 2:

\[ \text{Total arrangements with 2 boys between girls} = 2 \times 11! \]

### Step 5: Total Arrangements with Restrictions
Now, we need to find the total number of arrangements where there are at least 3 boys between the two girls. We can do this by subtracting the cases where the girls are together, with 1 boy between them, and with 2 boys between them from the total arrangements:

\[ \text{Required arrangements} = 14! - (2 \times 13! + 2 \times 12! + 2 \times 11!) \]

### Step 6: Factor Out \(12!\)
Now, we can factor out \(12!\) from the equation:

\[ \text{Required arrangements} = 12! \left( 14 \times 13 - (2 \times 13 + 2 \times 12 + 2 \times 11) \right) \]

### Step 7: Simplifying the Expression
Calculating the terms inside the parentheses:

1. \(14 \times 13 = 182\)
2. \(2 \times 13 = 26\)
3. \(2 \times 12 = 24\)
4. \(2 \times 11 = 22\)

Adding these:

\[ 26 + 24 + 22 = 72 \]

Now substituting back:

\[ \text{Required arrangements} = 12! \left( 182 - 72 \right) = 12! \times 110 \]

### Conclusion
Thus, the number of ways to arrange 12 boys and 2 girls such that there are at least 3 boys between the two girls is given by:

\[ \text{Required arrangements} = \lambda \times 12! \]

Where \(\lambda = 110\).

### Final Answer
The value of \(\lambda\) is **110**.

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