Binomial Theorem
The Pascal’s Triangle is an arrangement of numbers that is generated by summing up adjacent numbers to from the next row.
Construction of the Pascal’s Triangle
Before we can go onto understanding the uses of the triangle we need to first be able to construct it. Firstly, we need to know that the first row of Pascal’s Triangle is row zero and that this row only contains one ‘1’. Now imagine that at the start and end of each row, there is an imaginary 0.
If we were to sum two adjacent numbers in the first row, we will obtain the next row. This would mean that row one would contain two ‘1’s as seen in the diagram below. We can observe that each row has n + 1 entries and the first and last number of each row is 1.
Since Pascal’s triangle scales infinitely, we can repeat this process as many times as we want to until row infinity. We also denote the rth entry of the nth row with,
Replacing the numbers, with their respective binomial coefficients,
Binomial Coefficients
Besides being part of the Pascal’s Triangle, The binomial coefficient is equal to
The binomial coefficient is also equivalent to the number of ways to choose r objects from n objects. For example, if you wanted to pick 5 people from a group of 20 people, the number of ways you could do that would be:
The binomial coefficients in the Pascal’s Triangle are read out loud as ‘n choose r’. Other commonly notations for the binomial coefficients include, nCr, C(n,r). We can conclude that if, 0< r < n, where n is a natural number then,
- Intuitively, the sum of two numbers in the Pascal’s Triangle is equal to the number below them. Algebraically,
Another property of the binomial coefficient is,
- On the Pascal’s Triangle, this tells us that the first and last number in a row is always 1. Combinatorically, this is equal to the number of ways you can choose 0 objects from n objects or n objects from n objects. There is only one way you can choose no objects (which is just to not choose any objects!) and one way you can choose all the objects (which is just to choose all the objects!). Using the formula,
Binomial Theorem
We can observe that the coefficients of the binomial expansions are identical to the values in the binomial expansions.
From this we can say that the values in the Pascal’s Triangle can be used to find the coefficients of binomial expansions such as (x + y)ⁿ.For example, to find the expansion of (x + y)³, we sub in the values of n into the binomial theorem,
We can think about this combinatorically. How many possible strings containing x,y and of length 3, have exactly two x’s in them? The strings are: xxy, xyx, yxx. This is because when ( x+ y )³ is expanded completely, each term in the expansion corresponds to exactly one such string. To determine how many such strings there are, we need to determine how many ways there are to place the ‘y’. Since in a string of length 3, there are, by definition, 3C1 possible positions to place 1 ‘y’s (y__, _y_, __y). In other words, the coefficient of x²y = 3C1 = 3.
Generalizing,
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