An ice skating competition features 8 skaters. How many different ways can the skaters finish the competition? How many different ways can 3 of the skaters finish first, second and third?
8P8 = 8! = 40320
8P3 = 8*7*6 = 336
8P8 = 8! = 40320
8P3 = 8*7*6 = 336
To determine the number of different ways the skaters can finish the competition, we need to use the concept of permutations.
1. To find the number of ways the 8 skaters can finish the competition, we use the formula for permutations of a set:
- The number of possibilities for the first place is 8 (all 8 skaters can be in the first place).
- The number of possibilities for the second place is 7 (since one skater has already taken the first place).
- The number of possibilities for the third place is 6 (since two skaters have already taken the first and second places).
- Similarly, the number of possibilities for each subsequent place decreases by 1.
- Therefore, the total number of different ways the skaters can finish the competition is 8 * 7 * 6 * ... * 3 * 2 * 1, which can be written as 8!.
2. To find the number of ways 3 skaters can finish first, second, and third, we can again use the concept of permutations.
- The number of possibilities for the first place is 8 (all 8 skaters can be in the first place).
- The number of possibilities for the second place is 7 (since one skater has already taken the first place).
- The number of possibilities for the third place is 6 (since two skaters have already taken the first and second places).
- Therefore, the number of different ways 3 skaters can finish first, second, and third is 8 * 7 * 6, which can be written as 8P3.
Now, let's calculate the answers:
1. The number of different ways the skaters can finish the competition is 8! (factorial).
Calculating 8!:
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40,320
2. The number of different ways 3 skaters can finish first, second, and third is 8P3 (permutation of 8, taken 3 at a time).
Calculating 8P3:
8P3 = 8 * 7 * 6 = 336
Therefore, the skaters can finish the competition in 40,320 different ways, and 3 skaters can finish first, second, and third in 336 different ways.