To determine the number of groups of three players that are all juniors, we need to recognize that we are dealing with combination problems. Combinations are used to determine how many ways we can select a certain number of items from a larger group, where the order of selection does not matter.
Given these terms:
- [tex]\({ }_{14} C _6\)[/tex]
- [tex]\({ }_{14} C _3\)[/tex]
- [tex]\({ }_6 C _3\)[/tex]
- 20
- 364
- 3,003
We need to identify the correct combinations that result in the numbers given by the provided solution and match them with the correct interpretations.
### Calculation for Total Combinations from 14 Players
The combination notation [tex]\({ }_{14} C _3\)[/tex] represents the number of ways to choose 3 players out of a group of 14. Using the formula for combinations:
[tex]\[
{ }_{14} C _3 = \frac{14!}{3!(14-3)!}
\][/tex]
According to the result, this calculation should give us 364.
### Calculation for Junior Combinations from 6 Juniors
The combination notation [tex]\({ }_6 C _3\)[/tex] represents the number of ways to choose 3 players out of a group of 6 juniors. Using the formula for combinations:
[tex]\[
{ }_6 C _3 = \frac{6!}{3!(6-3)!}
\][/tex]
According to the result, this calculation should give us 20.
### Matching the Numerical Values
From the six given terms, two of them reflect our calculations:
- [tex]\({ }_{14} C _3\)[/tex] contributes to the total combinations of any players from the 14-player pool, which is 364.
- [tex]\({ }_6 C _3\)[/tex] contributes to the combinations of selecting 3 players all of whom are juniors from the 6 juniors, which is 20.
Thus, the two terms that represent the number of groups of three players that are all juniors are:
- 20
- [tex]\({ }_6 C _3\)[/tex]
