At a birthday party, 3 girls and 3 boys are seated in a row. How many different arrangements are possible if the first seat is taken by a boy, the last seat is taken by a girl, and John is not seated next to each of the other boys?

To solve this problem, we will break it down step by step and calculate the number of valid seating arrangements under the given conditions.

### Total number of arrangements without restrictions
There are 3 boys and 3 girls. The problem specifies that a boy must sit in the first seat, and a girl must sit in the last seat. 

1. **Assign the first and last seats:**
   - The first seat must be taken by one of the 3 boys, so there are 3 choices for the first seat.
   - The last seat must be taken by one of the 3 girls, so there are 3 choices for the last seat.

2. **Arrange the remaining 4 people:**
   After placing a boy in the first seat and a girl in the last seat, we are left with 2 boys and 2 girls to arrange in the remaining 4 seats. The total number of ways to arrange these 4 people is \(4!\) (since we are arranging 4 distinct people).

   \[
   4! = 24
   \]

Thus, the total number of seating arrangements without any restrictions is:

\[
3 \times 3 \times 4! = 3 \times 3 \times 24 = 216
\]

### Restriction: John should not sit next to any of the other boys
Now, we impose the restriction that John, one of the 3 boys, must not be seated next to any of the other boys. To account for this restriction, we will use the inclusion-exclusion principle.

1. **Case 1: John is seated next to one of the other boys**
   - First, we treat John and one of the other boys as a "block." This means we are effectively treating the block as one unit. Since John can sit next to 2 other boys, we have 2 ways to form the block.
   - Now, instead of arranging 6 people, we are arranging 5 units (the "John + boy" block, 1 other boy, and 3 girls). The number of ways to arrange these 5 units is \(5!\).
   - Inside the block, John and the other boy can be seated in 2 different ways (John can be either on the left or on the right).

   Therefore, the number of ways to seat John next to one of the other boys is:

   \[
   2 \times 5! \times 2 = 2 \times 120 \times 2 = 480
   \]

2. **Case 2: John is seated next to both other boys**
   - In this case, we treat all 3 boys as a single "block." There is only 1 way to form this block.
   - Now, we are arranging 4 units (the "3-boy block" and 3 girls), so the number of ways to arrange these 4 units is \(4!\).
   - Inside the block, the 3 boys can be seated in \(3!\) different ways.

   Therefore, the number of ways to seat all 3 boys together is:

   \[
   1 \times 4! \times 3! = 1 \times 24 \times 6 = 144
   \]

3. **Apply inclusion-exclusion principle**
   Using inclusion-exclusion, the number of ways to seat John next to one or more boys is:

   \[
   480 - 144 = 336
   \]

### Final calculation
The total number of valid seating arrangements where John is not seated next to any of the other boys is:

\[
216 - 336 = 120
\]

Thus, there are **120 different arrangements** possible.

Would you like more details on any step?

Here are 5 related questions to consider:
1. What if there were 4 boys and 4 girls instead?
2. How does the inclusion-exclusion principle apply to larger groups?
3. What if there were no restriction about John not sitting next to other boys?
4. How many arrangements are possible if John must sit in the middle seat?
5. How many arrangements are possible if the boys and girls must alternate seats?

**Tip:** The inclusion-exclusion principle helps account for overlapping restrictions when calculating combinations or arrangements.

Explore the combinatorics problem of arranging 3 boys and 3 girls in a row with specific restrictions. Learn how to apply permutation and the inclusion-exclusion principle to find the total number of valid seating arrangements. Suitable for students in Grades 10-12.

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