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1. In how many different ways can a first, second, and third prize be awarded in a game with eight contestants?
- To determine the number of ways the prizes can be awarded, we can use the concept of permutations.
- Since the order matters (i.e., first, second, third), we can use the formula for permutations.
- The formula for permutations of selecting r items from n items is given by: nPr = n! / (n-r)!
- In this case, we have 8 contestants for the first prize, 7 remaining contestants for the second prize, and 6 remaining contestants for the third prize.
2. Formula:
- The formula to calculate permutations is nPr = n! / (n-r)!
- n = total number of contestants
- r = number of prizes to be awarded (in this case, 3: first, second, third)
- ! denotes factorial, which means the product of all positive integers up to that number.
3. Solution:
- Substituting the values into the formula, we get:
- For first prize: 8P1 = 8! / (8-1)! = 8! / 7! = 8 ways
- For second prize: 7P1 = 7! / (7-1)! = 7! / 6! = 7 ways
- For third prize: 6P1 = 6! / (6-1)! = 6! / 5! = 6 ways
Therefore, the total number of ways the first, second, and third prizes can be awarded in the game with eight contestants is 8 * 7 * 6 = 336 ways. Each prize is awarded by selecting a different contestant each time.
