Question 6 options:
Use the five-step thinking plan to help you solve this problem.
Thirty-six students are to be divided into teams of equal size. How many different ways can the students be divided (assuming each team must have more than 1 member)?
1. Understanding the problem: The problem is asking for the number of different ways that 36 students can be divided into teams of equal size, with each team having more than 1 member.
2. Analyzing the problem: To solve this problem, we need to consider the factors of 36 and calculate the number of possible team sizes.
3. Creating a plan: We will list the factors of 36 and calculate the possible team sizes.
4. Solving the problem:
The factors of 36 are:
1, 2, 3, 4, 6, 9, 12, 18, 36
However, each team must have more than 1 member, so we eliminate the factor 1.
Possible team sizes: 2, 3, 4, 6, 9, 12, 18, 36
5. Checking the answer: We have calculated all the possible team sizes based on the factors of 36.
Step-by-step solution:
1. Understand the problem: The problem states that there are thirty-six students that need to be divided into teams of equal size. The teams must have more than one member. We need to determine the number of different ways the students can be divided.
2. Plan: We will use the five-step thinking plan to solve this problem.
3. Execute the plan:
a. Step 1: Identify the goal - Determine the number of different ways to divide the students.
b. Step 2: Gather information - There are thirty-six students.
c. Step 3: Formulate a strategy - We can use a systematic approach to count the number of ways to divide the students.
d. Step 4: Execute the strategy - Start by considering the possible team sizes, starting from 2 members and going up to half of the total number of students (18 members).
- If we divide the students into teams of 2, there are 36/2 = 18 teams.
- If we divide the students into teams of 3, there are 36/3 = 12 teams.
- If we divide the students into teams of 4, there are 36/4 = 9 teams.
- If we divide the students into teams of 5, there are 36/5 = 7 teams.
- If we divide the students into teams of 6, there are 36/6 = 6 teams.
- If we divide the students into teams of 7, there are 36/7 = 5 teams.
- If we divide the students into teams of 8, there are 36/8 = 4 teams.
- If we divide the students into teams of 9, there are 36/9 = 4 teams.
- If we divide the students into teams of 10, there are 36/10 = 3 teams.
- If we divide the students into teams of 11, there are 36/11 = 3 teams.
- If we divide the students into teams of 12, there are 36/12 = 3 teams.
- If we divide the students into teams of 13, there are 36/13 = 2 teams.
- If we divide the students into teams of 14, there are 36/14 = 2 teams.
- If we divide the students into teams of 15, there are 36/15 = 2 teams.
- If we divide the students into teams of 16, there are 36/16 = 2 teams.
- If we divide the students into teams of 17, there are 36/17 = 2 teams.
- If we divide the students into teams of 18, there are 36/18 = 2 teams.
e. Step 5: Evaluate the results - Add up the number of teams for each team size:
- 18 + 12 + 9 + 7 + 6 + 5 + 4 + 4 + 3 + 3 + 3 + 2 + 2 + 2 + 2 + 2 = 94
So, there are 94 different ways to divide the students into teams of equal size.
4. Review the solution: The problem asked for the number of different ways to divide the students into teams of equal size, assuming each team must have more than one member. The solution obtained using the five-step thinking plan is 94.
To solve this problem, we can follow the five-step thinking plan:
Step 1: Understand the problem.
We want to divide 36 students into teams of equal size, with each team having more than 1 member. We need to determine the number of different ways we can do this.
Step 2: Devise a plan.
We can first find all the factors of 36, which will give us the possible team sizes. Then, we can calculate the number of ways to divide the students based on each team size.
Step 3: Carry out the plan.
Let's find the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, and 36. We can disregard 1 and 36 since they represent a team with only one member. We are left with the following team size options: 2, 3, 4, 6, 9, 12, 18.
For a team size of 2: The number of ways to select teams is C(36, 2) = 36! / ((2!(36-2)!) = 36! / (2!34!) = (36 * 35) / (2 * 1) = 630.
For a team size of 3: The number of ways to select teams is C(36, 3) = 36! / ((3!(36-3)!) = 36! / (3!33!) = (36 * 35 * 34) / (3 * 2 * 1) = 7140.
We can repeat this process for the remaining team sizes.
Step 4: Evaluate the solution.
By calculating the number of ways to divide the students for each team size, we can determine the total number of different ways.
Step 5: Reflect on the solution.
Using the five-step thinking plan helped us break down the problem into manageable steps, allowing us to efficiently find the answer.