Sure, let's break down how to determine which two terms represent the number of groups of three players that are all juniors.
When we talk about forming groups, we often use combinations, which are represented by [tex]\({ }_{n} C _r\)[/tex], where [tex]\( n \)[/tex] is the total number of items to choose from, and [tex]\( r \)[/tex] is the number of items to choose. The combination formula is given by:
[tex]\[ { }_{n} C _r = \frac{n!}{r!(n-r)!} \][/tex]
Firstly, let's evaluate [tex]${ }_{14} C _3$[/tex].
[tex]\[
{ }_{14} C _3 = \frac{14!}{3!(14-3)!} = \frac{14 \times 13 \times 12}{3 \times 2 \times 1} = 364
\][/tex]
Secondly, let's evaluate [tex]${ }_{6} C _3$[/tex].
[tex]\[
{ }_{6} C _3 = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20
\][/tex]
From these calculations, we see that [tex]\({ }_{14} C _3 = 364\)[/tex] and [tex]\({ }_{6} C _3 = 20\)[/tex], which are the two numbers representing the number of groups of three players that are all juniors. Thus, the two terms are 364 and 20.
