To solve the problem of dividing 20 people into 2 teams of 10 people each, follow these steps:
### Step-by-Step Solution
1. Determine the Total Number of People:
There are 20 people.
2. Choose 10 People for the First Team:
To form a team of 10 out of 20 people, you need to calculate the number of combinations. The formula for combinations is given by:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\][/tex]
Where [tex]\( n \)[/tex] is the total number of items, and [tex]\( k \)[/tex] is the number of items to choose. In our scenario, [tex]\( n = 20 \)[/tex] and [tex]\( k = 10 \)[/tex].
3. Calculate the Number of Ways to Choose 10 People Out of 20:
The number of combinations for choosing 10 people out of 20 is:
[tex]\[
\binom{20}{10} = \frac{20!}{10! \cdot 10!}
\][/tex]
Using the combination formula, you get:
[tex]\[
\binom{20}{10} = 184756
\][/tex]
This means there are [tex]\( 184756 \)[/tex] ways to choose 10 people out of 20.
4. Avoid Double Counting:
Since dividing remaining 10 people from the remaining 20 automatically forms the other team, dividing the same groups in reverse order (Team A then B, or Team B then A) counts as one unique division. Thus, we must divide the total number of ways by 2 to avoid double counting:
[tex]\[
\text{Total ways} = \frac{184756}{2} = 92378
\][/tex]
### Conclusion
Therefore, the number of ways to divide 20 people into 2 teams of 10 people each is:
[tex]\[
92378
\][/tex]
So, the answer is [tex]\( 92378 \)[/tex] ways.
