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.

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 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.

Explanation:

Analyze part (c) counting technique

Using the Permutations knowledge point

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

The order of the arrangement matters (bottom, middle, top), so permutations are used.

Analyze part (d) counting technique

Using the Combinations knowledge point

$$ C(21, 3) = \frac{21!}{3!(21-3)!} = \frac{21 \times 20 \times 19}{3 \times 2 \times 1} = 1330 $$

The order of the arrangement does not matter, so combinations are used.

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.

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

Question 2

• 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.

You can create <blank>1330</blank> different triple cones without repeating a flavor.