Tag: Counting Principles

This will be the final post in the series on counting principles. Here, I wish to deal with a strange idea of balancing parentheses. If that seems like a strange term itself, let me explain. An arrangement of n left parentheses ‘(‘ and n right parentheses ‘)’ is said to be balanced if, for every…

Launching Pad We are in the middle of a series on counting principles. In The Inclusion-Exclusion Principle, we saw a method of counting that ensured we counted everything while not double counting anything. Now, we are ready to go ahead with our study of derangements as promised. Understanding Derangements Let us first define what a…

Recapitulation We are currently in the middle of a series on counting principles. At the conclusion of the previous post, I had promised you that we would look at the idea of derangements. However, I realized that we need to lay some ground work before we get to derangements. When we count, how do we ensure…

After an extended break for Lent, we continue with the series on counting principles. I had ended the last post in the series with a promise that we will look into the technique of partitioning. And I had left you with a problem to solve: Given that x, y, and z are natural numbers, how many solutions exist to…

The Fundamental Theorem of Arithmetic We started a series on counting principles earlier this year. We first considered the most basic principles, namely the multiplication and addition principles. Then we looked at how permutations differ from combinations, considering some basic patterns of both including the elements of Pascal’s triangle. Next, we turned our attention to…

Recapitulation In this post we continue with the series on Counting Principles. Whereas we devoted the previous post to permutations of special kinds, we will turn our attention today to the matter of combinations. In Arranging and Choosing we saw that the binomial coefficients are related to the elements of Pascal’s triangle. We saw that the…

Refresher Two weeks back, we started a new series on Counting Principles. In Down for the Count we looked the the basic multiplication and addition principles of counting. In Arranging and Choosing we introduced the ideas of permutations and combinations. In this post, we will consider circular permutations and permutations with restrictions. Circular Permutations When we…

Defining Terms In the previous post, Down for the Count, we started a series on counting principles. We looked at the multiplication and addition principles of counting and I promised that, in this post, we would look at permutations and combinations, as well as the relationship between combinations and the binomial coefficients. So, what do these…

What Are Counting Principles? Counting is something that we learn from a very young age, either in the informal environment at home or in the formal environment of school. It forms the basis of all the mathematics we learn through our lives. All the basic mathematical operations (addition, subtraction, multiplication, division, and exponentiation) can be…