How Many Ways Can a Team of 11 Be Selected from 22 People?

The problem of determining the number of ways to select a team of 11 from 22 people can be solved using the combination formula.A combination is a selection of items where the order does not matter. This is often denoted as ( binom{n}{k} ), where ( n ) is the total number of items, and ( k ) is the number of items to be chosen.

Understanding the Combination Formula

The combination formula is given by:

[ binom{n}{r} frac{n!}{r!(n - r)!} ]

where n is the total number of people to choose from, and r is the number of people to choose.

Applying the Formula to Our Problem

In this scenario, we need to select 11 people from a group of 22. Therefore, we will use:

[ binom{22}{11} frac{22!}{11! cdot 22 - 11!} frac{22!}{11! cdot 11!} ]

Step-by-Step Calculation

Let's break down the factorial calculations:

[ 22! 22 times 21 times 20 times 19 times 18 times 17 times 16 times 15 times 14 times 13 times 12 times 11! ]

Therefore, we only need to calculate the numerator and the denominator separately:

Numerator Calculation

[ 22 times 21 462 ]

[ 462 times 20 9240 ]

[ 9240 times 19 175560 ]

[ 175560 times 18 3160080 ]

[ 3160080 times 17 53722560 ]

[ 53722560 times 16 859560960 ]

[ 859560960 times 15 12893414400 ]

[ 12893414400 times 14 180518817600 ]

[ 180518817600 times 13 2346744628800 ]

[ 2346744628800 times 12 28160935545600 ]

Denominator Calculation

[ 11! 39916800 ]

Final Calculation

[ binom{22}{11} frac{28160935545600}{39916800} 705432 ]

Thus, the number of ways to select a team of 11 from 22 is 705432.

Examples and Variations

Selecting 11 men from 20 men:

The formula becomes:

[ binom{20}{11} frac{20!}{11!(20 - 11)!} frac{20!}{11! cdot 9!} 167960 ]

Therefore, you can select 11 men from 20 in 167,960 different ways.

Selecting 11 members when 2 are already chosen:

Here, you need to choose 9 more players from the remaining 20 - 2 18 players.

The formula is:

[ binom{18}{9} frac{18!}{9!(18 - 9)!} frac{18!}{9! cdot 9!} 48620 ]

This indicates that you can choose the remaining 9 players from 18 in 48,620 different ways.

Conclusion

The combination formula is a powerful tool for solving problems related to team selection and has various applications in mathematics, statistics, and real-world scenarios. By using this formula, you can determine the exact number of ways to form a team from a larger pool of candidates.