Key Concepts and Formulas

• Permutations: The number of ways to arrange $n$ distinct objects in a sequence is $n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1$.
• Factorial: $n!$ (n factorial) represents the product of all positive integers less than or equal to $n$.
• Complementary Counting: Sometimes, it's easier to calculate the total number of possibilities and subtract the number of undesirable possibilities to find the number of desirable possibilities.

Step-by-Step Solution

Step 1: Arrange the blocks of girls and boys

Since all the girls must stand together and all the boys must stand together, we can consider the group of girls as one block (G) and the group of boys as another block (B). We need to arrange these two blocks in a queue. The possible arrangements are GB or BG. The number of ways to arrange these two blocks is: $2! = 2$

Step 2: Arrange the girls within their block

There are 3 girls, and they can be arranged among themselves in $3!$ ways. $3! = 3 \times 2 \times 1 = 6$

Step 3: Arrange the boys without restrictions

There are 4 boys. If there were no restrictions, they could be arranged in $4!$ ways. $4! = 4 \times 3 \times 2 \times 1 = 24$

Step 4: Calculate the number of arrangements where $B_1$ and $B_2$ are adjacent

We need to find the number of arrangements where boys $B_1$ and $B_2$ are adjacent. We treat $B_1$ and $B_2$ as a single unit. Within this unit, $B_1$ and $B_2$ can be arranged in $2!$ ways ($B_1B_2$ or $B_2B_1$). Now we have this unit and the other 2 boys, making a total of 3 units to arrange. These 3 units can be arranged in $3!$ ways. Therefore, the number of arrangements where $B_1$ and $B_2$ are adjacent is: $2! \times 3! = 2 \times (3 \times 2 \times 1) = 2 \times 6 = 12$

Step 5: Calculate the number of arrangements where $B_1$ and $B_2$ are not adjacent

To find the number of arrangements where $B_1$ and $B_2$ are not adjacent, we subtract the number of arrangements where they are adjacent (calculated in Step 4) from the total number of possible arrangements of the boys (calculated in Step 3). $4! - (2! \times 3!) = 24 - 12 = 12$

Step 6: Calculate the total number of valid arrangements

Now we multiply the number of ways to arrange the blocks (Step 1), the number of ways to arrange the girls (Step 2), and the number of ways to arrange the boys such that $B_1$ and $B_2$ are not adjacent (Step 5). $\text{Total ways} = 2 \times 6 \times 12 = 144$

Whoops, there's an error. The given correct answer is 120. Let's recalculate.

Step 1: Arrange the blocks of girls and boys

Since all the girls must stand together and all the boys must stand together, we can consider the group of girls as one block (G) and the group of boys as another block (B). We need to arrange these two blocks in a queue. The possible arrangements are GB or BG. The number of ways to arrange these two blocks is: $2! = 2$

Step 2: Arrange the girls within their block

There are 3 girls, and they can be arranged among themselves in $3!$ ways. $3! = 3 \times 2 \times 1 = 6$

Step 3: Arrange the boys without restrictions

There are 4 boys. If there were no restrictions, they could be arranged in $4!$ ways. $4! = 4 \times 3 \times 2 \times 1 = 24$

Step 4: Calculate the number of arrangements where $B_1$ and $B_2$ are adjacent

We need to find the number of arrangements where boys $B_1$ and $B_2$ are adjacent. We treat $B_1$ and $B_2$ as a single unit. Within this unit, $B_1$ and $B_2$ can be arranged in $2!$ ways ($B_1B_2$ or $B_2B_1$). Now we have this unit and the other 2 boys, making a total of 3 units to arrange. These 3 units can be arranged in $3!$ ways. Therefore, the number of arrangements where $B_1$ and $B_2$ are adjacent is: $2! \times 3! = 2 \times (3 \times 2 \times 1) = 2 \times 6 = 12$

Step 5: Calculate the number of arrangements where $B_1$ and $B_2$ are not adjacent

To find the number of arrangements where $B_1$ and $B_2$ are not adjacent, we subtract the number of arrangements where they are adjacent (calculated in Step 4) from the total number of possible arrangements of the boys (calculated in Step 3). $4! - (2! \times 3!) = 24 - 12 = 12$

Step 6: We need to consider the positions.

The girls and boys can switch positions in two ways. The girls can be arranged in 3! ways. The boys can be arranged in such a way that $B_1$ and $B_2$ are not together in $4! - 2!3! = 24 - 12 = 12$ ways. We also need to consider the case where the two blocks switch position, we multiply by 2. Then the total number of ways is $2 \times 3! \times (4! - 2!3!) = 2 \times 6 \times 12 = 144$.

Oops, still incorrect. The correct answer is 120. Let's revisit the initial setup. There appears to be an error in the question or the provided answer. The logic and calculations are correct, and the steps lead to 144. The problem is mathematically sound. Let's consider an edge case. If we had only one boy, the answer would have to be 0. If we had zero boys, the answer would have to be $2 * 6 * 1 = 12$. It looks like there is an error with the given answer.

Let's try to get 120. $2 \times 5 \times 12 = 120$? Where would 5 come from? $2 \times 6 \times 10 = 120$? Where would 10 come from? It seems impossible to arrive at 120 with the given conditions.

Since all steps have been carefully reviewed and the logic is sound, it's highly probable the provided correct answer is wrong.

Common Mistakes & Tips

• Carefully consider the restrictions and use complementary counting when it simplifies the problem.
• Remember to account for arrangements within blocks when objects are grouped together.
• Double-check all calculations to avoid arithmetic errors.

Summary

We first arranged the blocks of girls and boys. Then, we arranged the girls within their block. Next, we calculated the total number of ways to arrange the boys and subtracted the number of arrangements where $B_1$ and $B_2$ are adjacent to find the number of arrangements where they are not adjacent. Finally, we multiplied all these values together to find the total number of valid arrangements. However, the result is 144, not the stated 120, indicating an error in the question or the provided answer.

Final Answer The final answer is \boxed{144}. The solution obtained through logical steps does not match any of the options, but it is most likely that the correct answer is actually 144, which does not correspond to any of the options. There is likely an error in the question or the given "correct answer" is wrong.