A specialty bakery allows customers to create a custom dessert box. Each box must contain … — ACT

A specialty bakery allows customers to create a custom dessert box. Each box must contain $1$ base, $1$ flavor, and $2$ different toppings. Customers can choose from $3$ base options, $4$ flavor options, and $6$ topping options. How many different custom dessert boxes can a customer create?

A
$72$
B
$144$
$180$ Answer
D
$216$
E
$360$

Answer

180

different custom dessert boxes can be created.

The total number of outcomes is found by multiplying the number of choices for each component. For the base, there are

3

choices. For the flavor, there are

4

choices. For the

2

different toppings chosen from

6

, order does not matter, so we calculate the combination

\binom{6}{2} = 15

. Multiplying these choices together gives

3 \times 4 \times 15 = 180

Step-by-Step Solution

Determine the number of ways to choose the base and flavor.

There are

3

base options and

4

flavor options.

Each selection is an independent choice.

Calculate the number of ways to choose

2

different toppings out of

6

available toppings using combinations.

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

The order in which the toppings are selected does not matter, so we use the combination formula

\binom{n}{k} = \frac{n!}{k!(n-k)!}

Apply the Fundamental Counting Principle to find the total number of unique dessert boxes.

Total combinations =

3 \times 4 \times 15 = 180

Multiply the number of ways to make each independent choice.

Key Concept

Fundamental Counting Principle and Combinations

Estimated Time:1m 15s

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