The Binomial Theorem: Connecting the Pieces of a Puzzle
The binomial theorem describes the algebraic expansion of powers of a binomial. But first, what is a binomial? Well, it is simply a two-term polynomial or, in other words, an algebraic expression having exactly two unlike terms, including the variables and the constant. According to the theorem, it possible to expand the polynomial (x + y)n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers, with b + c = n, and a (the coefficient) is a specific positive integer depending on n and b.
Who Came up with it?
The binomial theorem was discovered in 1665 by Isaac Newton, although the general form of the theorem (for any real number n), and a proof was published in 1736 by John Colson. It is important to mention, though, that special cases of the binomial theorem were known since at least the 4th century BC when Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent n=2. To add on, Greek mathematician Diophantus went a step further and cubed various binomials. Similarly, Indian mathematician Aryabhata's method for finding cube roots, suggest that he also knew the binomial formula for exponent n=3.
Moreover, binomial coefficients (values of a) which are combinatorial quantities, (that is to say, expressing the number of ways of selecting k objects out of n without replacement) were of particular interest to ancient Indian mathematicians. Indian lyricist Pingala (c. 200 BC) wrote the earliest known reference to this combinatorial problem in his Chandaḥśāstra, which contains a method for its solution. And by the 6th century AD, Indian mathematicians most likely knew how to express such a problem as the following quotient:
But first, let us backtrack for a second and explain what this all means. What do combinatorial problems have to do with the binomial theorem? To answer this we must introduce something called Pascal's Triangle.
Pascal's Triangle is constructed in the following way: in the topmost row there is a unique nonzero entry of 1. Each entry of each subsequent row is created by adding the number above and to the left with the number above as well as the right, treating blank entries as 0 (that is why each row starts and ends with 1). Now, I am sure if you remember learning about the binomial theorem before, you probably recall memorising the first five rows of Pascal's Triangle as a way of attaining the binomial coefficients quickly and efficiently.
But how exactly does this connect to combinatorial problems? Well, we can generalise Pascal's Triangle by stating: the kth entry in the nth row of Pascal's triangle can be denoted:
For example:
This is otherwise known as a combination. A combination is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter. For example, the number of combinations you can make with the letters A, B and C is only one. (I.e. there is only one possible way to combine the three letters, the order does not matter: we can write ABC, BAC, CBA, etc. and still have the same combination of three letters. Permutations, on the other hand, deal with arrangements, where the order does in fact matter.) The formula for combinations can be expressed as follows:
And relating back to Pascal's triangle:
The cool thing about this is that Pascal's triangle determines the coefficients which arise in binomial expansions. Let us prove this by expanding (x + y)2
The coefficients, as you can see, are the entries in the second row of Pascal's triangle:
1, 2, 1.
Proof of the Binomial Theorem
Just like any mathematical theorem, the Binomial Theorem requires a proof to show that it is applicable to all values of n. There are several ways of going about this, but I only plan to explain one of them.
COMBINATORIAL PROOF
When you expand (x + y)n, every term in the expansion is a product of n factors, where each factor is either x or y. So, each product comes from choosing either x or y from each of the n binomial factors.
This gives you a total of 2n products, since for each of the n factors, there are 2 choices (x or y).
Each product in the expansion can be written as xn-kyk, where k is the number of times y appears, and n-k is the number of times x appears, Since each factor is either x or y, for each product, you have exactly k factors of y and n-k factors of x.
The next goal is to count how many terms in the expansion result in a product of the form xn-kyk . This is equivalent to counting how many ways we can choose exactly k positions for y out of the n positions available.
This is a combinatorics problem: how many ways can you choose k positions for y in a sequence of length n? The answer is given by the number of ways to choose k elements from an n-element set, which is:
So, the number of terms equal to xn-kyk in the expansion is exactly n choose k.
The number (n choose k) can also be interpreted in a different way, as the number of distinct sequences of length n that can be formed using x and y, where exactly k positions are occupied by y and the remaining n-k positions are occupied by x.
So, by this reasoning, (n choose k) represents both:
- The number of terms in the expansion of (x + y)n that are of the form xn-kyk .
- The number of ways to place k y's in a sequence of n terms, with the remaining spots being x's.
Given this, the binomial expansion of (x + y)n can be written as:
Generalization of the Binomial Expansion
Consider the geometric series
where x is not equal to 0.
For which values of x does this series converge? Convergence, for those that are unfamiliar with the term, is when adding the terms of series, one after the other, in the order given by the indices, gives us a partial sum closer and closer to a given number. For example:
Going back to the question, what is the sum to infinity for this series when it converges?
You can write the answers to these two questions as follows:
In other words:
If you want to expand (1 -x)-2 you could say that it is equivalent to ((1 -x)-1)2 = (1 -x)-1(1 -x)-1
...
This was the method Newton used to generalize the binomial theorem to allow for negative and fractional exponents. Based on his findings we can conclude:
Real-Life Applications of the Binomial Theorem
While researching about the binomial theorem, I asked myself, what is this used for anyway? What are the real-world applications of it?
Well, the answer to this question is actually quite surprising. It is used in fields like insurance and finance to evaluate risk by estimating the probabilities of different outcomes over multiple trials. Additionally, it is applicable to quantitative finance, binomial models are used to calculate option prices (e.g., the binomial options pricing model), helping investors understand the potential future value of options over time. When analyzing the stresses on beams or other load-bearing structures, engineers use binomial expansions to approximate complex load distributions. Binomial expansions are also used to predict the probability of different genotypes arising from a set of parents with particular gene. In game theory, particularly in competitive and cooperative games, binomial expansions help calculate probabilities and optimize strategies by analyzing possible outcomes over multiple turns or stages.
All in all, the binomial theorem is a lot more complex than what we see on the surface. Both the history behind its discovery as well as its practical uses in different fields certainly surprised me. The fact that combinatorial formulas were a well-known fact to Indian mathematicians long before Isaac Newton's time make me think about how knowledge surpasses time and borders. It is fascinating to realize that the principles of the binomial theorem, something we often take for granted, have been studied and applied for centuries across different cultures, each mathematician adding on to the knowledge of their predecessor.
