Question
There are 15 students competing in a science fair. The top 5 students will be awarded the following ribbons: 1st - blue; 2nd - red; 3rd - yellow; 4th - white; and 5th - pink.
How many ways can the medals be awarded?
How many ways can the medals be awarded?
Answers
GPT-4o mini
To determine how many ways the top 5 students can be awarded different ribbons (blue for 1st, red for 2nd, yellow for 3rd, white for 4th, and pink for 5th), we can use the concept of permutations since the order matters.
We start with 15 students. The number of ways to choose and arrange the top 5 students can be calculated as follows:
1. **Select the 1st place student**: There are 15 options.
2. **Select the 2nd place student**: After selecting the 1st place student, there are 14 remaining options.
3. **Select the 3rd place student**: After selecting the 1st and 2nd place students, there are 13 remaining options.
4. **Select the 4th place student**: After selecting the 1st, 2nd, and 3rd place students, there are 12 remaining options.
5. **Select the 5th place student**: After selecting the 1st, 2nd, 3rd, and 4th place students, there are 11 remaining options.
Thus, the total number of ways to award the ribbons can be computed as:
\[
15 \times 14 \times 13 \times 12 \times 11
\]
Calculating this step-by-step:
1. \( 15 \times 14 = 210 \)
2. \( 210 \times 13 = 2730 \)
3. \( 2730 \times 12 = 32760 \)
4. \( 32760 \times 11 = 360360 \)
Therefore, the total number of ways the medals can be awarded is:
\[
\boxed{360360}
\]
We start with 15 students. The number of ways to choose and arrange the top 5 students can be calculated as follows:
1. **Select the 1st place student**: There are 15 options.
2. **Select the 2nd place student**: After selecting the 1st place student, there are 14 remaining options.
3. **Select the 3rd place student**: After selecting the 1st and 2nd place students, there are 13 remaining options.
4. **Select the 4th place student**: After selecting the 1st, 2nd, and 3rd place students, there are 12 remaining options.
5. **Select the 5th place student**: After selecting the 1st, 2nd, 3rd, and 4th place students, there are 11 remaining options.
Thus, the total number of ways to award the ribbons can be computed as:
\[
15 \times 14 \times 13 \times 12 \times 11
\]
Calculating this step-by-step:
1. \( 15 \times 14 = 210 \)
2. \( 210 \times 13 = 2730 \)
3. \( 2730 \times 12 = 32760 \)
4. \( 32760 \times 11 = 360360 \)
Therefore, the total number of ways the medals can be awarded is:
\[
\boxed{360360}
\]