A high school debate team must elect the president, vice president, and treasurer from 8 to 12 members. How many ways can they be chosen Jesus?
To determine the number of ways they can be chosen, we need to calculate the number of permutations (order matters) for each position.
For the president position, there are 8 possible candidates.
For the vice president position, there are 7 remaining candidates after one is chosen for the president position.
For the treasurer position, there are 6 remaining candidates after two have been chosen for the president and vice president positions.
The total number of ways to choose the president, vice president, and treasurer is found by multiplying the number of candidates for each position:
8 * 7 * 6 = 336.
Therefore, there are 336 different ways they can be chosen.
To find the number of ways the president, vice president, and treasurer can be chosen from a group of 8 to 12 members, we will calculate the number of combinations for each possible group size.
1. When the group size is 8:
- The president can be chosen in 8 ways.
- After the president is chosen, the vice president can be chosen in 7 ways (as the same person cannot hold both positions).
- Finally, the treasurer can be chosen in 6 ways (since the same person cannot hold all three positions).
- The total number of ways to choose the president, vice president, and treasurer from 8 members is 8 * 7 * 6 = 336 ways.
2. When the group size is 9:
- Following the same process, the number of ways to choose the president, vice president, and treasurer from 9 members is 9 * 8 * 7 = 504 ways.
3. When the group size is 10:
- Again following the process, the number of ways to choose the president, vice president, and treasurer from 10 members is 10 * 9 * 8 = 720 ways.
4. When the group size is 11:
- The number of ways to choose the president, vice president, and treasurer from 11 members is 11 * 10 * 9 = 990 ways.
5. When the group size is 12:
- The number of ways to choose the president, vice president, and treasurer from 12 members is 12 * 11 * 10 = 1320 ways.
Therefore, the high school debate team can be chosen in a total of 336 + 504 + 720 + 990 + 1320 = 3870 ways.