A club consists of 16 men and 19 women. In how many ways can they choose a president, vice president, treasurer, and secretary, along with an advisory committee of six people? (Round the answer to five decimal places.)
_____ x 10(to the 11th power) ways
For the first 4 positions, the order matters, so we have a permutation, followed by a combination
so number of ways
= 35x34x33x32xC(31,6)
= 9.2524 x 10^11
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A school is electing students for class officers. There are four offices: president, vice president, secretary, and treasurer. The student with the most votes becomes president, second highest number of votes is vice president, and so on. If there are students in this class, how many ways can students be elected?
To find the total number of ways the club can choose a president, vice president, treasurer, and secretary, along with an advisory committee of six people, we need to calculate the number of choices for each role and then multiply them together.
1. First, let's determine the number of choices for each role:
- President: There are 35 members in total (16 men + 19 women), so there are 35 possible choices for the president.
- Vice president: After choosing the president, there will be 34 remaining members, so there are 34 choices for the vice president.
- Treasurer: After choosing the president and vice president, there will be 33 remaining members, so there are 33 choices for the treasurer.
- Secretary: After choosing the president, vice president, and treasurer, there will be 32 remaining members, so there are 32 choices for the secretary.
2. Next, let's calculate the number of choices for the advisory committee:
- The advisory committee consists of 6 people.
- We need to choose 6 people from the remaining 31 members (after choosing the president, vice president, treasurer, and secretary).
- The number of ways to choose 6 people from 31 can be calculated using the combination formula: nCr = n! / r!(n-r)!
- So, the number of choices for the advisory committee is C(31, 6).
3. Finally, let's calculate the total number of ways the club can choose the president, vice president, treasurer, secretary, and advisory committee:
- Multiply the number of choices for each role: 35 × 34 × 33 × 32 = 1,256,640.
- Multiply the number of choices for the advisory committee: 1,256,640 × C(31, 6).
Calculating C(31, 6):
- C(31, 6) = 31! / (6! × (31-6)!)
- Using a calculator or math software, we find that C(31, 6) = 593,775.
The total number of ways the club can choose a president, vice president, treasurer, secretary, and advisory committee is 1,256,640 × 593,775 = 746,560,400.
Rounding this answer to five decimal places, we get 746.56040 × 10^6.
Therefore, the answer is 7.46560 × 10^8 ways.