7. Principle of Inclusion Exclusion (PIE): problem set

Problem 7.5.1

How many positive numbers less than or equal to 100 are multiples of 2 or 3?

Problem 7.5.2

How many positive numbers less than or equal to 100 are NOT multiples of 2 or 3?

Problem 7.5.3

Out of 200 students, there are 100 taking Geometry, 70 taking Algebra, and 30 taking both. How many students are taking neither?

Problem 7.5.4

AMC 8

Sets $A$ and $B$ , shown in the Venn diagram, have the same number of elements. Their union has 2007 elements and their intersection has 1001 elements. Find the number of elements in $A$ .

Problem 7.5.5

In a town of 351 adults, every adult owns a car, motorcycle, or both. If 331 adults own cars and 45 adults own motorcycles, how many of the car owners do not own a motorcycle?

Problem 7.5.6

AMC 8

At Euler Middle School, 198 students voted on two issues in a school referendum with the following results: 149 voted in favor of the first issue and 119 voted in favor of the second issue. If there were exactly 29 students who voted against both issues, how many students voted in favor of both issues?

Problem 7.5.7

How many positive numbers less than or equal to 200 are multiples of 2 and 3 but not a multiple of 5?

Problem 7.5.8

AMC 8

Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren?

Problem 7.5.9

AMC 10

Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of 5 chairs under these conditions?

Problem 7.5.10

AIME

Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement is equally likely, the probability that such a license plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left to right as it does right to left) is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Problem 7.5.11(additional)

AMC 10

There are 20 students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are 10 students taking yoga, 13 taking bridge, and 9 taking painting. There are 9 students taking at least two classes. How many students are taking all three classes?

Problem 7.5.12(additional)

BmMT

Students are being assigned to faculty mentors in the Berkeley Math Department. If there are 7 distinct students and 3 distinct mentors, and each student has exactly one mentor, in how many ways can students be assigned to mentors such that each mentor has at least one student?

Problem 7.5.13(additional)

AMC 12

Call a number prime-looking if it is composite but not divisible by 2, 3, or 5. The three smallest prime-looking numbers are 49, 77, and 91. There are 168 prime numbers less than 1000. How many prime-looking numbers are there less than 1000?

Problem 7.5.14(additional)

MATHCOUNTS

How many different subsets of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$ contain at least one element in common with each of the sets $\{2, 4, 6, 8, 10, 12\}$ , $\{3, 6, 9, 12\}$ and $\{2, 3, 5, 7, 11\}$ ?

Principle of Inclusion Exclusion (PIE): all problems