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

1. Step 1: Consider the volunteers as a group of 20 distinct individuals.

2. Step 2: Choose 5 volunteers out of 20 to form the first team. There are $\binom{20}{5}$ ways to do this.

3. 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 $\binom{15}{5}$ ways.

4. Step 4: For the third team, we choose 5 volunteers from the remaining 10, which can be done in $\binom{10}{5}$ ways.

5. Step 5: The last 5 volunteers will automatically form the fourth team, so there is only 1 way to choose them.

6. 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 $4! = 24$.

The formula to calculate the total number of ways to divide the volunteers is therefore: $\frac{\binom{20}{5} \times \binom{15}{5} \times \binom{10}{5} \times \binom{5}{5}}{4!}$

Calculations

1. Calculate each binomial coefficient:

   • $\binom{20}{5} = \frac{20!}{5!(20-5)!} = 15,504$
   • $\binom{15}{5} = \frac{15!}{5!(15-5)!} = 3,003$
   • $\binom{10}{5} = \frac{10!}{5!(10-5)!} = 252$
   • $\binom{5}{5} = 1$

2. Multiply these values together and divide by 24: $\frac{15,504 \times 3,003 \times 252 \times 1}{24} = 4,845,363,600$

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:

1. How many ways are there to divide 12 people into 3 teams of 4 each?
2. If the teams were labeled (e.g., Team A, Team B), how would the solution change?
3. What if the teams were of different sizes, say 10, 5, and 5?
4. Can you apply a similar approach to divide 18 people into 3 teams of 6 each?
5. 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.