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. how many different sundaes can you create using one of the ice cream flavors and one of the toppings? what counting technique will you use to make this calculation?

a. combinations because the selections come from a single group of items and the order of the arrangement does not matter.
b. the multiplication principle because there are m possible outcomes for one group and n outcomes for the other group.
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 126 different sundaes.
(type a whole number.)

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.

Explanation:

Identify the counting technique for part a

Using the Fundamental Counting Principle knowledge point

• To find the number of sundaes with one flavor (out of 21) and one topping (out of 6), we choose one item from each of two distinct groups.
• The multiplication principle applies because there are \(M = 21\) outcomes for the first group and \(N = 6\) outcomes for the second group.
• This matches option B.

Calculate the number of sundaes for part a

Using the Fundamental Counting Principle knowledge point

$$ \text{Total sundaes} = 21 \times 6 = 126 $$

Identify the counting technique for part b

Using the Arrangements with Repetitions knowledge point

• We are creating a triple cone with 3 positions (bottom, middle, top) from \(n = 21\) flavors.
• Since the same flavor can be used more than once and the order (bottom, middle, top) matters, we are making \(r = 3\) selections from \(n = 21\) choices with repetition.
• This matches option D.

Calculate the number of triple cones for part b

Using the Arrangements with Repetitions knowledge point

$$ \text{Total triple cones} = 21^3 = 9261 $$

Answer:

Question 1

• (A) Combinations because the selections come from a single group of items and the order of the arrangement does not matter.
• (B) The multiplication principle because there are M possible outcomes for one group and N outcomes for the other group. (Correct answer)
• (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 <blank>126</blank> different sundaes.

Question 2

• (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.