Math Problem Statement
Solution
To solve this problem, we need to calculate the number of ways to divide 20 volunteers into 4 teams, each consisting of 5 volunteers.
Solution
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Step 1: Consider the volunteers as a group of 20 distinct individuals.
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Step 2: Choose 5 volunteers out of 20 to form the first team. There are ways to do this.
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Step 3: After selecting the first team, 15 volunteers remain. We then choose another 5 volunteers out of these 15 for the second team, which can be done in ways.
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Step 4: For the third team, we choose 5 volunteers from the remaining 10, which can be done in ways.
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Step 5: The last 5 volunteers will automatically form the fourth team, so there is only 1 way to choose them.
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Step 6: Since the teams are not labeled (the order of teams doesn’t matter), we must divide by the number of ways to arrange 4 identical teams, which is .
The formula to calculate the total number of ways to divide the volunteers is therefore:
Calculations
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Calculate each binomial coefficient:
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Multiply these values together and divide by 24:
Answer
There are 4,845,363,600 different ways to divide the 20 volunteers into 4 teams of 5.
Would you like further details or have any questions?
Here are some related questions for further practice:
- How many ways are there to divide 12 people into 3 teams of 4 each?
- If the teams were labeled (e.g., Team A, Team B), how would the solution change?
- What if the teams were of different sizes, say 10, 5, and 5?
- Can you apply a similar approach to divide 18 people into 3 teams of 6 each?
- How does the answer change if there were only 15 volunteers to be divided into 3 teams of 5?
Tip: When dividing people or items into groups where the groups are indistinguishable, remember to divide by the factorial of the number of groups to account for identical arrangements.
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Math Problem Analysis
Mathematical Concepts
Combinatorics
Combination Counting
Indistinguishable Groups
Formulas
Combination formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)
Division of identical arrangements: \( \frac{\text{Total arrangements}}{k!} \) where \( k \) is the number of indistinguishable groups
Theorems
Basic Counting Principle
Suitable Grade Level
Grades 10-12