Question

A mathematics student goes out for ice cream to celebrate an A on her midterm. She decides on an ice creams sundae. The available flavors are Chocolate, Strawberry and Vanilla. The available toppings are Hot fudge, Caramel, Nuts, Sprinkles and Strawberries. (a) How many sundaes are possible if one flavor is selected and three different toppings are selected? (b) How many sundaes are possible if there are three scoops of ice cream each which may be of any flavor and two distinct toppings are selected?

Answer 1

(a) 30

(b) 10

Explanation:

There 3 different flavors of ice-cream available and 5 different toppings are available.

In mathematics, the procedure to select k items from n distinct items, without replacement, is known as combinations.

The formula to compute the combinations of k items from n is given by the formula:

`{n\choose k}=(n!)/(k!\cdot (n-k)!)`

(a)

Compute the number of possible sundaes that can be made with one flavor and three different toppings as follows:

`\#\text{Sundaes}=\#\text{Ways to select 1 flavor}* \#\text{Ways to select 3 toppings}`

`={3\choose 1}* {5\choose 3}\\\\=3* 10\\\\=30`

(b)

Compute the number of possible sundaes that can be made if there are three scoops of each ice cream and two different toppings as follows:

`\#\text{Sundaes}=\#\text{Ways to select 1 flavor}* \#\text{Ways to select 3 toppings}`

`=1* {5\choose 2}\\\\=1* 10\\\\=10`