To determine how many ways you can select a group of 6 friends from a group of 22 friends, you'll use the concept of combinations. Combinations are used when you want to choose a subset of items from a larger set where the order does not matter.
The formula for combinations is given by:
[tex]\[ _nC_r = \frac{n!}{r!(n-r)!} \][/tex]
where:
- [tex]\( n \)[/tex] is the total number of items (in this case, friends),
- [tex]\( r \)[/tex] is the number of items to choose,
- [tex]\( n! \)[/tex] (n factorial) is the product of all positive integers up to [tex]\( n \)[/tex],
- [tex]\( r! \)[/tex] (r factorial) is the product of all positive integers up to [tex]\( r \)[/tex],
- [tex]\((n-r)!\)[/tex] is the factorial of the difference between [tex]\( n \)[/tex] and [tex]\( r \)[/tex].
For this problem:
- [tex]\( n = 22 \)[/tex] (total number of friends),
- [tex]\( r = 6 \)[/tex] (number of friends to select).
Plugging these values into the combination formula gives:
[tex]\[ _{22}C_6 = \frac{22!}{6!(22-6)!} = \frac{22!}{6! \cdot 16!} \][/tex]
To simplify the computation, note that the factorials can be expanded, but a direct approach is not always practical; instead, let's follow through with the steps to better understand conceptually:
1. Calculate [tex]\( 22! \)[/tex], the factorial of 22.
2. Calculate [tex]\( 6! \)[/tex], the factorial of 6.
3. Calculate [tex]\( 16! \)[/tex], the factorial of 16.
4. Divide [tex]\( 22! \)[/tex] by the product of [tex]\( 6! \)[/tex] and [tex]\( 16! \)[/tex].
However, using the formula directly might not be feasible for manual calculation due to the large numbers involved, but this is the theoretical understanding.
From these steps or a computational approach, we find that the number of ways to select 6 friends from 22 friends is:
[tex]\[ 74613 \][/tex]
Thus, there are 74,613 ways to choose a group of 6 friends from a group of 22 friends.
