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 7980 different triple cones without repeating a flavor.
(type a whole number.)

d. using the 21 flavors, how many different triple cones can you create with 3 different flavors if you dont care about the order of the flavors of the cone? what counting technique will you use to make this calculation?

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

you can create different triple cones when order does not matter.
(type a whole number.)

Explanation:

Identify the counting scenario

Using the Combinations concept, we analyze the selection of 3 different flavors from 21 available flavors. Since we do not care about the order of the flavors on the cone, the order of selection does not matter.

Select the appropriate technique

Because the selections come from a single group of items (21 flavors) and the order of the arrangement does not matter, the correct counting technique is combinations. This corresponds to option B.

Calculate the number of combinations

Using the Combinations formula to choose \(r = 3\) items from \(n = 21\) items:

$$ C(21, 3) = \frac{21!}{3!(21-3)!} = \frac{21 \times 20 \times 19}{3 \times 2 \times 1} $$

Simplify the calculation

We compute the value:

$$ C(21, 3) = \frac{7980}{6} = 1330 $$

Answer:

Question 1

Part 1

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

Part 2

You can create <blank>1330</blank> different triple cones when order does not matter.