Binomial theorem

We are interested in finding relation between many things. When (a + b) is raised to different powers then relation between previous expansion with the next expansion has a very interesting result. This post is about this and binomial theorem. It is called binomial as it has two variables in it. In this post I will show you how we can arrive at the coefficients of the Binomial Expansion. The foundation of binomial theorem lies in Pascals Triangle. Pascals triangle is an interesting topic and I will cover it in detail. A Pascals Triangle looks like this.

                              1
                            1   1
                          1   2   1
                        1   3   3   1
                      1   4   6   4   1
                    1   5   10  10  5   1
                      so on.  

A pascals triangle is generated from coefficients of the expansion of different powers raised to (a + b). Let us expand some of the powers.

(a + b)1 = (a + b)

(a + b)2 = (a2 + 2ab + b2)

(a + b)3 = (a3 + 3a2b + 3ab2 + b3)

(a + b)4 = (a4 + 4a3b + 6a2b2 + 4ab3 + b4)

(a + b)5 = (a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5)

Let us express the coefficients in other form:

1   1
1   2   1
1   3   3   1
1   4   6   4   1
1   5   10  10  5   1
1   6   15  20  15  6  1
so on

Some points to note

1. The first column is a constant.
2. The second column is an arithmetic sequence whose common difference is 1. i.e. the terms are represented by n/1.
3. The third column needs some explanation
   The sequence is 1 3 6 10 15...
   The difference between successive terms is 2 3 4 5....
   The difference of elements again is 1 1 1.
   Don't worry about the method which I am going to follow. Learn the method. If on finding successive differences you reach at a constant in 2 steps. Suppose the sequence is formed by the second power. Here, the equation is an² + bn + c. Where n is the position of term. Let us take the first three terms of the sequence. Then we get a system of simultaneous equations.
   
   a + b + c = 1
   4a + 2b + c = 3
   9a + 3b + c = 6
   solving we get, a = 1/2, b = 1/2 and c = 0
   The equation is (1/2)(n²+n). In the table the sequence start from row 2 so we will replace n by (n-1).
   Then the equation becomes (1/2)((n-1)² + (n-1)).
   Reducing into factors we get n(n-1)/2 = n(n-1)/(1x2)
   The above thing in combinations is represented as ⁿC₂.
4. Following the above process and assuming the sequence depends on third power and solving and replacing n by (n-2) we get ⁿC₃.
5. Following the above process and assuming the sequence depends on fourth power and solving and replacing n by (n-3) we get ⁿC₄.
6. Following the above process we get ⁿC₅.
7. Following the above process we get ⁿC₆.

Hence the expansion of (a+b) raised to n^th power is

The binomial theorem
(a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁a^n-1b + ⁿC₂a^n-2b² + ... + ⁿC_n-1ab^n-1 + ⁿC_nbⁿ

Probability and Statistics

Probability and Statistics is one of the main branch of mathematics. It deals with population and chance. In statistics we read about stats of different kinds their arrangement, collection, classification, analysis and interpretation. In probability we deal with the study of chances and their ways. Probability mainly consists of permutations and combinations. There are many uses of probability in statistics. Let us have a look at what we study in probability and statistics.

Statistics has two different meanings. One is the data and other is a discipline. The data is facts and figures which we come across different conditions in our life and discipline refers to the collection, analysis and interpretation of those data. In the beginning of the study of statistics we deal with its different types of distributions. Then we deal with measures of central tendency. They are mean, median and mode. After it we deal with measures of dispersion. They are range, mean deviation and standard deviation. This study gives us a broad view of the data. The mean is the value which is the most probable when the whole data is taken into account. Median tells us about the middle, quartile, decile and percentile. Mode tells us which value occurs the most. The measure of dispersion tells us about the trend of the data, how much it is away with respect to the mean and the square of the mean. Then we deal with skewness and kurtosis. A measure of skewness would tell us how far the frequency curve of the given frequency distribution deviates from a symmetric one. On the other hand, a measure of kurtosis gives us some information about the degree of flatness (or peakedness) of the frequency curve. Then we deal with how one type of data is related to other type of data. Such study is dealt in correlation and regression. These things deals with statistics. To study more in statistics we need knowledge of probability.

In probability we deal with Random Experiment, sample space, event and algebra of events. Then we deal with probability of discrete sample space. After it we study about probability distributions. These are Bernoulli, Binomial, Multinomial, Hypergeometric, Geometric, Poisson and Negative Binomial. These are standard probability distributions. Then we deal with univariate distributions. After it we deal with standard continuous distributions. They are Normal, Gamma or Exponential, Beta Distributions. Then we deal with bivariate distributions. Then comes functions of random variables chi-square distribution, t-distribution and F-distribution. Then we end the topic by studying the methods used in factories.