To solve this problem, we need to determine the correct statement about the number of ways Liliana can choose 3 different appetizers from a list of 12 options.
The primary concept at play here is whether the order in which the appetizers are chosen matters or not.
1. Identifying if Order Matters:
- Combinations: If the order in which the appetizers are chosen does not matter, we use combinations.
- Permutations: If the order in which the appetizers are chosen does matter, we use permutations.
2. Clarifying the given situation:
- Liliana is going to serve 3 different appetizers from a list of 12 options.
- The problem states explicitly that the order does not matter.
Given this, we should use the combination formula:
[tex]\[ \text{Combinations} = \binom{n}{r} \][/tex]
where:
- [tex]\( n \)[/tex] is the total number of items.
- [tex]\( r \)[/tex] is the number of items to choose.
Here, [tex]\( n = 12 \)[/tex] and [tex]\( r = 3 \)[/tex].
Let's look at each statement provided:
A. There are [tex]\( 12 P_3 = 1,320 \)[/tex] ways Liliana can serve the appetizers because the order in which the appetizers are chosen doesn't matter.
- This statement is referring to permutations but mentions the order doesn't matter, which is contradictory.
B. There are [tex]\( \binom{12}{3} = 220 \)[/tex] ways Liliana can serve the appetizers because the order in which the appetizers are chosen doesn't matter.
- This statement accurately describes the scenario. It uses combinations and acknowledges that order does not matter.
C. There are [tex]\( 12 P_3 = 1,320 \)[/tex] ways Liliana can serve the appetizers because the order in which the appetizers are chosen matters.
- This statement indicates the use of permutations and correctly identifies that order matters. However, the problem states that order does not matter.
D. There are [tex]\( \binom{12}{3} = 220 \)[/tex] ways Liliana can serve the appetizers because the order in which the appetizers are chosen matters.
- This statement is contradictory; it correctly calculates the number of combinations but incorrectly states that order matters.
Based on our reasoning, the correct statement is:
B. There are [tex]\( \binom{12}{3} = 220 \)[/tex] ways Liliana can serve the appetizers because the order in which the appetizers are chosen doesn't matter.
