Question

1. I visit an ice cream place that has 60 flavors! Suppose I'm going to have a 4 scoop ice cream cone. I care about the order of flavors

a. How many choices do I have? (I can repeat the flavors)

b. I now want three different flavors. How many choices do I have?

c. I now decide that I only want one flavor for my 3 scoop cone. How many choices do I have?

d. I want two flavors for my 4 scoop cone. How many choices do I have?

e. In my ice cream cup it must include at least one scoop of chocolate chip, cookie dough, coffee, oreo. How many choices do I have if I have four different flavors? (Complements method)

f. I decided I wanted ice cream in a cup. How many choices with one flavor? How many chooses with two different flavors?  

g. I must have at least one scoop of mint chocolate chip, cookie dough, or coffee. How many choices do I have if I want four different flavors in my cup?

Answer 1

a.This is random sampling with replacment.

The total number of choices = No. of choices for 1st scoop * No. of choices for 2nd scoop * No. of choices for 3rd scoop * No. of choices for 4th scoop = 60*60*60*60 = 12960000

b.The total number of choices = No. of choices for 1st scoop * No. of choices for 2nd scoop * No. of choices for 3rd scoop = 60*60*60 = 216000

c.The 3 scoop cone should have the same flavor. Once the flavor for 1st scoop is chosen, the same flavor has to be used for 2nd cone and 3rd cone.

The total number of choices = No. of choices for 1st scoop * No. of choices for 2nd scoop * No. of choices for 3rd scoop = 60*1*1 = 60

d.Once the flavor for 1st scoop is chosen, the same flavor can not be chosen for 2nd scoop. Also the flavors of 3rd and 4th scoop are now fixed (same as 1st and 2nd scoop).

No. of choices for 1st scoop * No. of choices for 2nd scoop * No. of choices for 3rd scoop * No. of choices for 4th scoop = 60*59*1*1 = 3540

e.The 4 scoop cup should include one scoop of chocolate chip, cookie dough, coffee, oreo. Since the 4 flavor are decided,

No. of choices for 1st scoop * No. of choices for 2nd scoop * No. of choices for 3rd scoop * No. of choices for 4th scoop = 1*1*1*1 = 1

f.Number of choices with 1 flavor = 60

Number of choices with 2 different flavors = 60 * 59 = 3540

g.Need to have atleast 1 flavor of mint chocolate chip, cookie dough, or coffee.

Number of ways in which atleast 1 flavor of mint chocolate chip, cookie dough, or coffee is obtained = Total number of ways in which 60 flavors can be obtained - Number of ways in which none of the 3 flavors of mint chocolate chip, cookie dough, or coffee is obtained

= 60*60*60*60 - 57*57*57*57 = 2403999