A bakery allows customers to customize a fruit tart by selecting 1 crust type from 3 avail… — ACT

A bakery allows customers to customize a fruit tart by selecting $1$ crust type from $3$ available choices, $1$ filling type from $4$ available choices, and $2$ different fruit toppings from $5$ available choices. How many different fruit tart combinations are possible?

A
$17$
B
$60$
$120$ Cevap
D
$240$
E
$300$

Cevap

The total number of different fruit tart combinations possible is 120.

To find the total number of unique fruit tart combinations, apply the Fundamental Counting Principle across all decision stages. Selecting 1 crust out of 3 yields 3 options. Selecting 1 filling out of 4 yields 4 options. Selecting 2 different fruit toppings out of 5 requires choosing combinations without regard to order, which is calculated as 5 choose 2 = 10. Multiplying the options from each stage gives 3 * 4 * 10 = 120 total combinations.

Adım Adım Çözüm

Determine the number of ways to choose the crust and filling.

There are

3

crust choices and

4

filling choices.

Each selection stage is independent.

Calculate the number of ways to choose

2

different fruit toppings from

5

available options.

\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10

choices.

The order in which the two toppings are selected does not matter, so combinations must be used.

Apply the Fundamental Counting Principle to find the total combinations.

3 \times 4 \times 10 = 120

Multiply the number of choices for each independent step.

Anahtar Kavram

Fundamental Counting Principle and Combinations

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