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Answer :
To determine how many different ways a race with 6 runners can be completed, where there are no ties, we need to think about the number of possible rankings from 1st place to 6th place.
Let's break it down step by step:
1. Consider the 1st place:
There are 6 runners, so you have 6 choices for who can come in 1st place.
2. Consider the 2nd place:
Once the 1st place is decided, there are 5 runners left. So, you have 5 choices for who can come in 2nd place.
3. Continue for the remaining places:
- For the 3rd place, you have 4 remaining choices.
- For the 4th place, you have 3 choices.
- For the 5th place, you have 2 choices.
- Finally, for the 6th place, there is only 1 runner left, so there's just 1 choice.
To find the total number of different ways the race can be completed, you multiply these choices together:
[tex]\[ 6 \times 5 \times 4 \times 3 \times 2 \times 1 \][/tex]
Calculating this gives you:
- [tex]\(6 \times 5 = 30\)[/tex]
- [tex]\(30 \times 4 = 120\)[/tex]
- [tex]\(120 \times 3 = 360\)[/tex]
- [tex]\(360 \times 2 = 720\)[/tex]
- [tex]\(720 \times 1 = 720\)[/tex]
Therefore, there are 720 different ways the race can be completed with 6 runners.
Let's break it down step by step:
1. Consider the 1st place:
There are 6 runners, so you have 6 choices for who can come in 1st place.
2. Consider the 2nd place:
Once the 1st place is decided, there are 5 runners left. So, you have 5 choices for who can come in 2nd place.
3. Continue for the remaining places:
- For the 3rd place, you have 4 remaining choices.
- For the 4th place, you have 3 choices.
- For the 5th place, you have 2 choices.
- Finally, for the 6th place, there is only 1 runner left, so there's just 1 choice.
To find the total number of different ways the race can be completed, you multiply these choices together:
[tex]\[ 6 \times 5 \times 4 \times 3 \times 2 \times 1 \][/tex]
Calculating this gives you:
- [tex]\(6 \times 5 = 30\)[/tex]
- [tex]\(30 \times 4 = 120\)[/tex]
- [tex]\(120 \times 3 = 360\)[/tex]
- [tex]\(360 \times 2 = 720\)[/tex]
- [tex]\(720 \times 1 = 720\)[/tex]
Therefore, there are 720 different ways the race can be completed with 6 runners.
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