How many different ways can a teacher rearrange the students in the first row of their class that has 6 seats?
The correct answer and explanation is:
To determine the number of different ways a teacher can rearrange the students in the first row of 6 seats, we can calculate the number of possible permutations of the students. Since there are 6 students, each student must be assigned a specific seat. The number of arrangements can be found using the formula for permutations of distinct objects, which is: P(n)=n!P(n) = n!
Where nn is the total number of students, and n!n! (read as “n factorial”) represents the product of all positive integers from 1 to nn. In this case, there are 6 students, so we need to calculate 6!6!. 6!=6×5×4×3×2×1=7206! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
Therefore, there are 720 different ways to rearrange the 6 students in the first row of seats.
Explanation:
- First seat: For the first seat, the teacher can choose any one of the 6 students, so there are 6 options.
- Second seat: After seating the first student, there are 5 remaining students. Hence, there are 5 options for the second seat.
- Third seat: After seating two students, 4 students remain, giving 4 options for the third seat.
- Fourth seat: With 3 students left, there are 3 options for the fourth seat.
- Fifth seat: With 2 students remaining, there are 2 options for the fifth seat.
- Sixth seat: Finally, there is only 1 student left, so only 1 option for the sixth seat.
Multiplying these together (6 × 5 × 4 × 3 × 2 × 1) gives the total number of ways to arrange the students, which is 720.
This type of problem is an example of permutation, where the order of arrangement matters.