Question

Kimball’s Farm in Westford (MA) is known for its ice cream and mini golf. They have 50 flavors! Kiddo it going to have a 3 scoop ice cream cone. He cares about the order of the flavors (eg. perhaps vanilla won’t taste as good after a strong flavor like chocolate). You must show work in each question. (a) How many choices does kiddo have? Hint: He decides on the bottom flavor, and then on the middle flavor, and then on the top flavor. Flavors can be repeated. (b) Actually, he wants three different flavors. How many choices does he have for his ice cream cone? Mandatory: Use permutations. (c) And now he decides that he only wants one flavor for his 3 scoop cone. How many choices does he have for his ice cream cone? (d) Use the method of complements to find the number of ways he can choose a 3 scoop cone with exactly 2 flavors. Hint: The answers to the previous parts are 125000 and 117600 and 50 respectively. (e) Now kiddo wants ice cream in a bowl, so order of scoops no longer matters. He’ll have three different flavors. i. How many choices does he have for an ice cream bowl with 3 different flavors and 2 of 5 toppings? ii. How many choices does he have if he has a scoop of vanilla? Hint: What is the opposite of “have at least one scoop of vanilla”?

Answer 1

(a). he may choose different flavours for different scoops from 50 flavours

so no of choices are 50C1*50C1*50C1 = 125000

b). for choosing 3 different flavours no repeatation should be done

so 50C1*49C1*48C1 = 117600

c). no of outcomes of 3 scoops of same flavour is he choose one flavour from 50 and chooses no from remaining 49 that is

50C1*49C0*49C0 = 50

about d he have to choose any two flavours first, that is 50C1*49C1 = 2450
then he have to arrange these flavours in 3 different places i.e, (2C1*2C1*2C1) - 1 - 1 = 6 ways(three scoops of same colours is eliminated)
so joint selection is 2450*6 = 14700
for e1). he have to select 3 different flavours from 50 is 50C1*49C1*48C1 = 117600
and 2 toppings from 5 i.e, 5C2 = 10
total selection is 117600*10 = 1176000

for e2). he already have vanilla

so he have to select from remaining flavours i.e, 49C1*48C1 = 2352

and asusual 2 toppings of 5 i.e, 5C2 = 10

so no.of ways = 2352*10 = 23520