In order to determine the number of ways to select a group of 6 friends out of a group of 22 friends, we'll use the combination formula [tex]\( \binom{n}{r} \)[/tex]:
[tex]\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \][/tex]
Here, [tex]\( n = 22 \)[/tex] (the total number of friends) and [tex]\( r = 6 \)[/tex] (the number of friends to be selected). Let's calculate each part step-by-step.
1. Calculate [tex]\( 22! \)[/tex]:
[tex]\[
22! = 22 \times 21 \times 20 \times \ldots \times 2 \times 1 = 1124000727777607680000
\][/tex]
2. Calculate [tex]\( 6! \)[/tex]:
[tex]\[
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
\][/tex]
3. Calculate [tex]\( (22 - 6)! \)[/tex]:
[tex]\[
(22 - 6)! = 16! = 16 \times 15 \times 14 \times \ldots \times 2 \times 1 = 20922789888000
\][/tex]
4. Substitute these values into the combination formula:
[tex]\[
\binom{22}{6} = \frac{22!}{6!(22-6)!} = \frac{1124000727777607680000}{720 \times 20922789888000}
\][/tex]
5. Simplify the expression:
[tex]\[
\binom{22}{6} = \frac{1124000727777607680000}{15064808720640000} = 74613
\][/tex]
Therefore, the number of ways to select a group of 6 friends out of 22 friends is [tex]\( 74,613 \)[/tex].
