Problem of the Week Archive
Ice Cream Shoppe – June 26, 2017
Problems & Solutions
Small Town Ice Cream Shoppe carries five kinds of ice cream: vanilla, chocolate, strawberry, lemon
custard, and caramel swirl. They also have three types of toppings: sprinkles, chopped nuts, and hot
fudge. A “Shoppe Supreme” consists of three scoops of ice cream, not necessarily distinct flavors,
and three scoops of toppings, also not necessarily distinct. The ice cream is always scooped before
the toppings are added, in how many distinct ways can a Shoppe Supreme be assembled at Small
Town Ice Cream Shoppe?
For each of the three scoops of ice cream there are 5 options and for each of the three scoops of toppings there are 3 options.
In total, there are 5 × 5 × 5 × 3 × 3 × 3 =
3375
ways to assemble a Shoppe Supreme.
In the Ice Cream Shoppe, there are stools at the counter. Janet,
Mason, Renae, Steve and Trina are going to sit side-by-side at the
counter so that there are no empty seats between them. If Trina and
Mason must sit next to each other, in how many possible distinct
orders can they sit?
Since Trian and Mason must sit together, let’s consider them as one unit. That means there are 4! = 24 orders in which the four
“units” can sit. However, Trina and Mason could switch places within their unit, thus there are 24 × 2 =
48
orders in which the
five people can sit.
Finally, Janet, Mason, Renae, Steve and Trina’s orders are done but the waitress has forgotten who
ordered what (and she is too embarrassed to ask). If she randomly places the drinks down, one in
front of each person, what is the probability that she sets each of the five drinks down in front of the
person who ordered it? Express your answer as a common fraction.
There are 5! = 120 orders in which the drinks could be set down. Only one placement would be correct, thus there is a
1/120
chance that she would get it right.
