Let's solve the problem step-by-step to find the probability that you and your 3 friends will get to sit together in the same row in a classroom with 27 students and 4 desks per row.
1. Calculate the total number of students and rows:
- Total number of students in the class: [tex]\(27\)[/tex]
- Number of students including you and your 3 friends: [tex]\(4\)[/tex]
- Rows in the classroom: Since each row has 4 desks, the total number of rows is:
[tex]\[
\text{Rows} = \frac{27}{4} \approx 6.75
\][/tex]
Since you can't have a fraction of a row, it means not every row will be completely full. However, let's consider the exact seat allocation more clearly.
2. Determine the total number of possible seating arrangements:
- The total number of ways to arrange 27 students in 27 seats is given by [tex]\(27!\)[/tex] (27 factorial).
3. Calculate the number of favorable arrangements:
- Firstly, consider that you and your 3 friends need to be in the same row. The number of ways to arrange the 4 friends within one row is [tex]\(4!\)[/tex] (4 factorial).
- After seating you and your friends, the remaining seats are 23 (since 27 - 4 = 23). These 23 students can be arranged in 23! ways.
- The number of ways to choose a row for the 4 friends is equal to the number of rows, which is 7 (since dividing 27 by 4 gives approximately 6.75, thus rounding up to 7 rows).
4. Calculate the number of favorable arrangements:
[tex]\[
\text{Favorable arrangements} = 7 \times 4! \times 23!
\][/tex]
5. Compute the probability:
- The probability that you and your friends sit together in the same row is the ratio of the number of favorable arrangements to the total number of arrangements:
[tex]\[
\text{Probability} = \frac{\text{Favorable arrangements}}{\text{Total arrangements}} = \frac{7 \times 4! \times 23!}{27!}
\][/tex]
Simplifying this fraction:
[tex]\[
\text{Probability} = \frac{7 \times 24 \times 23!}{27!} = \frac{7 \times 24}{27 \times 26 \times 25}
\][/tex]
Do the multiplication and division:
[tex]\[
= \frac{7 \times 24}{17550} = \frac{168}{17550} \approx 0.0003418803418803419
\][/tex]
This matches one of the given answer choices. Thus, the probability that you and your 3 friends will get to sit together in the same row is:
[tex]\[
\boxed{\frac{1}{2925}}
\][/tex]
