Question

an ice cream shop offers 21 different flavors of ice cream and 6 different toppings. answer questions (a) through (d) using the appropriate counting technique (multiplication principle, arrangements with repetitions, permutations, or combinations). explain why you chose the particular technique.

a. the multiplication principle because there are m possible outcomes for one group and n outcomes for the other group.
b. combinations because the selections come from a single group of items and the order of the arrangement does not matter.
c. permutations because the selections come from a single group of items and the order of the arrangement matters.
d. arrangements with repetitions because there are r selections from a group of n choices and you can repeat choices.

you can create 9261 different triple cones.
(type a whole number.)

c. using the 21 flavors, how many different triple cones can you create with 3 different flavors if you specify which flavor goes on the bottom, middle and top? what counting technique will you use to make this calculation?

a. the multiplication principle because there are m possible outcomes for one group and n outcomes for the other group.
b. combinations because the selections come from a single group of items and the order of the arrangement does not matter.
c. permutations because the selections come from a single group of items and the order of the arrangement matters.
d. arrangements with repetitions because there are r selections from a group of n choices and you can repeat choices.

you can create different triple cones without repeating a flavor.
(type a whole number.)

Explanation:

Identify the counting technique

Using the Permutations knowledge point, we analyze the selection of 3 different flavors from 21 available flavors where the position (bottom, middle, top) matters. Since we are selecting from a single group of items without repetition and the order of the arrangement matters, the appropriate technique is permutations.

Select the correct explanation

We match the definition of permutations to the given options:

• Option A describes the multiplication principle for independent groups.
• Option B describes combinations (order does not matter).
• Option C describes permutations (order matters, single group).
• Option D describes arrangements with repetitions.

Thus, Option C is the correct choice.

Calculate the number of permutations

Using the Permutations formula for selecting \(r = 3\) items from \(n = 21\) distinct items:

$$ P(21, 3) = \frac{21!}{(21-3)!} = 21 \times 20 \times 19 $$

$$ P(21, 3) = 7980 $$

Answer:

Question 1

• (A) The multiplication principle because there are M possible outcomes for one group and N outcomes for the other group.
• (B) Combinations because the selections come from a single group of items and the order of the arrangement does not matter.
• (C) Permutations because the selections come from a single group of items and the order of the arrangement matters. (Correct answer)
• (D) Arrangements with repetitions because there are r selections from a group of n choices and you can repeat choices.

Question 2

You can create <blank>7980</blank> different triple cones without repeating a flavor.