How many ways can 13 students line up for lunch? This is a (permutation or combination.) 13 students can line can line up ways. (Do not answer in scientific notation.)
The Correct Answer and Explanation is:
o find the number of ways 13 students can line up, calculate the factorial of 13, which is written as 13!. This means multiplying all the whole numbers from 13 down to 1:13!=13×12×11×10×9×8×7×6×5×4×3×2×113! = 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 113!=13×12×11×10×9×8×7×6×5×4×3×2×1
Carrying out the multiplication:
- 13 × 12 = 156
- 156 × 11 = 1716
- 1716 × 10 = 17,160
- 17,160 × 9 = 154,440
- 154,440 × 8 = 1,235,520
- 1,235,520 × 7 = 8,648,640
- 8,648,640 × 6 = 51,891,840
- 51,891,840 × 5 = 259,459,200
- 259,459,200 × 4 = 1,037,836,800
- 1,037,836,800 × 3 = 3,113,510,400
- 3,113,510,400 × 2 = 6,227,020,800
- 6,227,020,800 × 1 = 6,227,020,800
Answer:
This is a permutation.
There are 6,227,020,800 ways for 13 students to line up.
Explanation:
A permutation problem arises when arranging distinct items in a sequence where the position of each item matters. If 13 students line up for lunch, the first person in line differs from the second, third, and so on. Swapping the positions of just two students results in a different arrangement. Since each student occupies a unique position, the total number of such distinct arrangements is calculated using factorial notation.
Using combination rules instead would ignore the order and only consider groups, which is not applicable here. In situations where arrangement or sequence matters, permutation is the correct concept. Hence, 13! gives the total number of possible sequences for 13 distinct students.
