MISCELLANEOUS COUNTING PROBLEMS
EXAMPLE 1.6.1
Refer to the Shakesperean Insult-o-Matic from MODULE 1.
Suppose we are going to form a three-part Shakespearean insult by choosing two different adjectives from column A and one noun phrase from column B.
Assume that rearranging the order of the adjectives does not change the meaning of the insult (for example, "You poisonous, dreadful toad" is the same as "You dreadful, poisonous toad").
How many different three-part insults are possible?
see solution
EXAMPLE 1.6.2
Suppose there are 8 Representatives and 6 Senators on a congressional conference committee. Three of each will be chosen to receive a birthday present, even though today isn't their birthday. How many 6-person groups are possible? (Note: they will each receive the same gift.)
see solution
EXAMPLE 1.6.3
There are 12 female aviators and 8 male aviators at Billy Bob's Blimp-o-Drome. Billy Bob is going to choose a pilot and co-pilot for a special women-only flight, and another pilot and co-pilot for a special men-only flight. How many combinations are possible?
see solution
EXAMPLE 1.6.4
Suppose that by show of hands, we find that 24 people in this room are 18 years old, and 20 people are 19 years old. If we ask the question, "How many are 18 or 19 years old?" will the answer be (24)(20) = 480?
FACTS
In counting problems or probability problems,
"AND" corresponds to "MULTIPLY"
"OR corresponds to "ADD"
EXAMPLE 1.6.5
A couple is expecting the birth of a baby. If the child is a girl, they will choose her first name and middle name from this list of their favorite girls' names: Betty, Beverly, Bernice, Bonita, Barbie.
If the child is a boy, they will choose his first name and middle name from this list of their favorite boys' names: Biff, Buzz, Barney, Bart, Buddy, Bert.
In either case, the child's first name will be different from the middle name. How many two-part names are possible?
A. 50
B. 600
C. 61
D. 900
see solution
EXAMPLE 1.6.6
The mathematics department is going to hire a new instructor. They want to hire somebody who possesses at least four of the following traits:
1. Honest;
2. Trustworthy;
3. Loyal;
4. Gets along well with others;
5. Good at math;
6. Good handwriting
In how many ways is it possible to combine at least four of these traits?
see solution
EXAMPLE 3.3.7
Six Democrats and five Republicans are in a jail cell. Either three Democrats or three Republicans will be chosen for a police line-up. How many different line-ups are possible? (We assume that a line-up is determined not only by the people who are chosen, but also by the order in which they are "lined-up.")
A. 990
B. 7200
C. 33
D. 180
EXAMPLE 1.6.8
Six Democrats and 5 Republicans are in a jail cell. Either 3 Democrats or 3 Republicans will be chosen to pick up trash alongside the highway. How many different 3-person groups are possible?
A. 30
B. 200
C. 7200
D. 180
see solution
EXAMPLE 1.6.9
Gomer's Auto Body Shoppe is still trying to unload some surplus paint by offering the following special deals:
For $99 you get a two-tone paint job, using one color for the car's top and a different color for the body.
For $129 you get a three-tone paint job, using one color for the car's top, a different color for the body, and a third color for the hub caps.
The available colors are:
1. hot pink
2. puke green
3. Gator orange
4. flaming chartreuse
How many different color schemes are possible?
A. 24
B. 36
C. 27
D. none of these
see solution
EXAMPLE 1.6.10
Gomer is going to order a frozen tofu cone. The following toppings are available:
granola crumbles
seaweed sprinkles
carob chips
frosted alfalfa sprouts
roasted soybeans
1. How many topping combinations are possible if he will choose either three or four toppings?
A. 50
B. 10
C. 15
D. 21
2. How many combinations are possible if he may choose all, some or none of the items?
see solution
EXAMPLE 1.6.11
At the animal shelter there are seven new puppies and six new kittens.
Either three puppies or three kittens will be randomly chosen and donated to a local nursing home. How many different three-animal collections are possible?
see solution
EXAMPLE 1.6.12
A survey of glamorous celebrities revealed that
30 wear dark glasses
32 smoke cigars
28 wear dark glasses and smoke cigars
1 does neither of these things
1. How many wear dark glasses and don't smoke cigars?
A. 540
B. 2
C. 35
D. 3
2. How many wear dark glasses or smoke cigars?
FACT
When a counting problem (or probability problem) refers to two categories that overlap (that is, categories that aren't mutually exclusive), the information can be organized with a Venn diagram.
see solution
There is another fact that can be used to answer question #2 above (but not question #1) without making the Venn diagram:
FACT:
For categories or conditions A, B:
n(A or B) = n(A) + n(B) - n(A and B)
Applying this fact to the question "How many wear dark glasses or smoke cigars?" we have:
n(wear dark glasses or smoke cigars)
= n(wear dark glasses) + n(smoke cigars) - n(wear dark glasses and smoke cigars)
= 30 + 32 - 28 = 34.
EXAMPLE 1.6.13
Among a group of Gators, 26 are modest, 28 are charming, 24 are modest and charming, 864 are not modest and not charming.
How many are...
1. ...charming but not modest?
2. ...modest or charming?
see solution
EXAMPLE 1.6.14
The editorial board of the National Requirer has decided to launch two new series of investigative reports. One series will involve the theme
"Elvis ____ my ____!"
For this series, the writers will generate a story line by filling in the first blank with an item from this list:
"loved"; "sang to"; "got sick on"; "smoked" and filling in the second blank with an item from this list:
"fig tree"; "living room carpet"; "cat".
The other series will involve the theme "Aliens ____ my ____!"
The writers will generate a story line by filling in the first blank with an item from this list: "attacked"; "rescued"; "probed";
and filling in the second blank with an item from this list:
"home town"; "trailer park"; "granny's arrest record"
How many story lines are possible?
see solution
EXAMPLE 1.6.15
Pizza Shack is offering the following special deal this month: for one low price ($6.99) you can get a large pizza with up to 4 different toppings. There are 8 toppings from which to choose. How many different topping combinations are possible?
A. 70
B. 1680
C. 163
D. 32
see solution
EXAMPLE 1.6.16
Ten points are distributed in a plane, in such a way that no three points are collinear. We will form either a triangle or a quadrilateral by selecting points to serve as vertices. How many different figures are possible?
see solution