The Binomial theorem is a part of elementary Algebra that explains the power of a binomial as the algebraic expression. Based on the theorem, we can expand the polynomial by adding up the values. The concept of binomial was written by early Mathematicians in 200 BC with solutions. At that time, the theorem was expressed in quotient format. Here, we will discuss different binomial formulas and related concepts for a better understanding of the topic and it will also help the students in the actual implementation of the theorem.

Binomial is Polynomial consists of two terms and the sum or differences of the monomial would always be a Binomial. The Binomial expression in mathematics is given as –

\[\ ax^{n} + bx^{m} \]

where a, b are the non- zero real coefficients and m, n are distinct non- negative integers.

Binomial Expansion is a method of expanding the expression of powers of a binomial term raised to any power. Or this is an Algebraic formula describing the algebraic expansion of a polynomial raised to different powers. For example, based on the binomial expansion theorem, you may expand the power of x + y into the sum form. The Binomial Expansion Formula in Mathematics is given as –

\[\ (x+y)^{n} = x^{n} + nx^{n-1}y + \frac{n(n-1)}{2!} x^{n-2} y^{2} + … + y^{n}\]

The binomial distributions are common in statistics and it has a maximum of two outputs for each binomial expression. For instance, if you toss a coin then the outcome may be either head or tail. Similarly, when you appear for a test then there are two possible occurrences either pass or fail. The binomial probability formula is used to calculate the success of the binomial distributions. There are the following properties associated with the concept.

- The number of trails is always fixed and given by small n.
- Each of the trails would be independent of other.
- The final output of each trail is fixed i.e. either pass or fail.
- The probability of success is given by the small p and its value always stays the same.
- The Binomial Probability Formula for exactly x number of successes and n number of trails is given by the Formula below –

\[\ P(X) = C_{x}^{n} P^{x} q^{n-x}\]

Where,

n = Total number of trials

x = Total number of successful trials

p = probability of success in a single trial

q = probability of failure in a single trial = 1-p

As the suggest binomial distribution can be taken as the common type of probability distribution with 2 possible outcomes. When we discuss the Probability Theory, the binomial distribution comes into two parameters i.e. n and p. The probability distribution becomes equal to the binomial probability distribution by satisfying the specific conditions. The formula for Binomial distribution in Mathematics is given below –

\[\ P(x) = \frac{n!}{r!(n-r)!} . p^{r}(1-p)^{n} = C(n, r).p^{r}(1-p)^{n-r}\]Where,

n = Total number of events

r = Total number of successful events.

p = Probability of success on a single trial._{n}C_{r} = \[\ \frac{n!}{r!(n − r)!} \]

n = Total number of events

r = Total number of successful events.

p = Probability of success on a single trial.

1 – p = Probability of failure.