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For her friend's birthday party, Liliana is going to serve 3 different appetizers from a list of 12 options. Which statement best describes this situation?

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.
B. There are [tex]${}_{12}C_3 = 220$[/tex] ways Liliana can serve the appetizers because the order in which the appetizers are chosen doesn't 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.
D. There are [tex]${}_{12}C_3 = 220$[/tex] ways Liliana can serve the appetizers because the order in which the appetizers are chosen matters.

Sagot :

GinnyAnsweridn
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.
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