Question

A group consists of 10 kids and 2 adults. On a hike, they must form a line with an adult at the front and an adult at the back. How many ways are there to form the line?

1) 12 2
2) 2-11!
3) 2-10!
4) 121

Answer 1

The total number of different ways to form the line with an adult at the front and back and 10 kids in the middle is 2 times 10! (option 3), because there are 2 ways to arrange the adults and 10! ways to arrange the kids in between them.

Step-by-step explanation:

The problem asks us to determine how many ways there are to form a line with a group on a hike where an adult must be at the front and the back of the line. With this constraint, the two adults have designated spots in the line-up, meaning there are 2 ways to arrange the adults (A1 at the front and A2 at the back, or A2 at the front and A1 at the back).

Once the adults are placed, the 10 kids can be arranged in any order between them. The number of distinct ways to arrange a given number of objects is given by the factorial of the number of objects, which is symbolized by an exclamation point (!). Therefore, the number of ways to arrange the 10 kids is 10! (which is equal to 10×9×8×7×6×5×4×3×2×1).

Since the arrangement of the adults is independent from the arrangement of the kids, we multiply the two possibilities together to get the total number of arrangements. So, the total number of ways to form the line is 2×10! (Option 3 is the correct answer).