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.

b. how many different triple cones can you create from the 21 flavors if the same flavor may be used more than once? assume that 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 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.

Explanation:

Analyze part (b) counting technique

Using the Arrangements with Repetitions knowledge point

$$ \text{Technique: Arrangements with repetitions} $$

$$ \text{Reason: } r \text{ selections from } n \text{ choices with repetition allowed and order matters.} $$

Calculate part (b) total arrangements

Using the Arrangements with Repetitions knowledge point

$$ n^r = 21^3 = 9261 $$

Analyze part (c) counting technique

Using the Permutations knowledge point

$$ \text{Technique: Permutations} $$

$$ \text{Reason: } r \text{ selections from } n \text{ choices without repetition and order matters.} $$

Calculate part (c) total permutations

Using the Permutations knowledge point

$$ P(21, 3) = \frac{21!}{(21-3)!} = 21 \times 20 \times 19 = 7980 $$

Answer:

Question b

• 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. (Correct answer)

You can create <blank>9261</blank> different triple cones.

Question c

• 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.

You can create <blank>7980</blank> different triple cones.