The Algebra and Combinatorics of PIE
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- The Principle of Inclusion-Exclusion (PIE) is a theorem that allows one to calculate the number of elements in a union of multiple finite sets. First, consider two sets, AAA and…
By Sujay Kazi, Victor Wang
- Here, rather than give a standard exercise or practice problem, I will explore some implications that PIE yields for positive real numbers. We first recall what PIE is: ∣A1∪A2∪.…
By Vincent Huang, Geir
Problems 6
- How many students like both apple and cherry pie?
- How many numbers between 1 and 100 (inclusive) are divisible by 2, 3, or 5? ("Or" includes the possibility that numbers are divisible by both 2 and 3, etc.)
- Every student in my school takes at least one foreign language class. The French teacher sees 100 of the students, and the Spanish teacher sees 150 of the students. There are 25…
- Among my friends, 10 people have brown hair and 12 people are left-handed. If each one of my friends has brown hair, is left-handed, or both, which of the following must definit…
- (2015 ARML Individual #10) Six people of different heights are getting in line to buy donuts. Compute the number of ways they can arrange themselves in line such that no three c…
- Justin the robot is on a mission to rescue abandoned treasure from a minefield. To do this, he must travel from the point (0,0,0)(0, 0, 0)(0,0,0) to (4,4,4)(4, 4, 4)(4,4,4) in t…