https://file.aiquickdraw.com/mathbot/file/1726135105331bo1aon3w.jpg

The problem asks how many ways a 3-person governing board (President, Treasurer, and Secretary) can be chosen from 30 club members, given that Alex and Jerry do not want to serve together on the board.

### Steps to solve the problem:

1. **Total number of ways to choose 3 people from 30 without restrictions:**
   Since the positions are distinct (President, Treasurer, Secretary), this is a permutation problem. The number of ways to choose and assign 3 people to 3 distinct positions from 30 members is given by:
   \[
   P(30, 3) = 30 \times 29 \times 28
   \]

2. **Number of ways Alex and Jerry serve together:**
   If Alex and Jerry are both on the board, we need to choose 1 more person for the remaining position. There are \(30 - 2 = 28\) members left to choose from for this position. The number of ways to assign Alex and Jerry to two distinct positions on the board (among President, Treasurer, Secretary) is \(P(3, 2) = 3 \times 2 = 6\). So, the total number of ways Alex and Jerry serve together is:
   \[
   6 \times 28 = 168
   \]

3. **Number of ways Alex and Jerry don’t serve together:**
   Subtract the number of ways Alex and Jerry serve together from the total number of ways to form the board:
   \[
   \text{Total ways without restrictions} - \text{Ways Alex and Jerry serve together} = 30 \times 29 \times 28 - 168
   \]

Now, let's calculate this.The number of ways to choose a 3-person governing board from 30 members, where Alex and Jerry do not serve together, is **24,192**.

Would you like further details or have any questions?

Here are 5 related questions to expand on this:
1. How would the answer change if the positions on the board were not distinct (i.e., no difference between President, Treasurer, and Secretary)?
2. What if the club had 40 members instead of 30?
3. How would the number of ways change if there were 4 positions instead of 3?
4. Can this method be applied to more complex scenarios with additional constraints (e.g., another pair who don’t want to serve together)?
5. How would we solve the problem if Alex and Jerry want to serve together, not apart?

**Tip:** When dealing with permutation problems involving constraints, break the problem into manageable steps by calculating both the unrestricted total and the restricted scenarios separately.

A club with 30 members wants to have a 3-person governing board (President, Treasurer, Secretary). In how many ways can this board be chosen if Alex and Jerry don’t want to serve together?

Learn how to calculate the number of ways to form a 3-person governing board from 30 club members, given that Alex and Jerry cannot serve together. This problem involves permutation formulas and combinatorics, suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation