A catering service allows customers to build a custom buffet meal by selecting 1 appetizer… — ACT

A catering service allows customers to build a custom buffet meal by selecting $1$ appetizer from $4$ available options, $1$ main course from $5$ available options, and $2$ distinct side dishes from $6$ available options. However, due to preparation constraints, $2$ specific side dishes cannot be selected together. How many different valid buffet meal combinations can a customer choose?

A
$23$
B
$120$
$280$ Answer
D
$300$
E
$600$

Answer

The total number of valid buffet meal combinations is

280

There are

4

appetizer choices and

5

main course choices. Choosing

2

side dishes out of

6

options gives

\binom{6}{2} = 15

total pairs. Subtracting the

1

disallowed pair leaves

14

valid side dish pairs. Multiplying these independent choices gives

4 \times 5 \times 14 = 280

valid buffet menu combinations.

Step-by-Step Solution

Calculate the total number of ways to select 2 distinct side dishes from 6 options.

\binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15

combinations.

Since the order in which the 2 side dishes are picked does not matter, combinations are used.

Subtract the restricted pair of side dishes from the total side dish combinations.

15 - 1 = 14

valid side dish combinations.

The problem specifies that 2 specific side dishes cannot be selected together, eliminating 1 specific pair.

Apply the Fundamental Counting Principle to find the overall number of valid buffet menu choices.

4 \text{ (appetizers)} \times 5 \text{ (main courses)} \times 14 \text{ (valid side pairs)} = 280

The choices for appetizer, main course, and side dish pair are independent, so their individual number of possibilities are multiplied.

Key Concept

Fundamental Counting Principle and Combinations with Restrictions

Estimated Time:1m 15s

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