IB SL Math AA-Binomial Expansion and Binomial Theorem

Factorial Notation :

To understand binomial theorem first you have to understand what the factorial notation is.

Factorial notation is a mathematical notation used to indicate the product of all positive integers up to and including a given positive integer.

The symbol for factorial is the exclamation mark "!".

For example, the factorial of 5 is 5! = 5 x 4 x 3 x 2 x 1 = 120. The factorial of a non-negative integer n is defined as: n! = n x (n-1) x (n-2) x ... x 2 x 1 with the special case that 0! = 1.

Binomial Coefficient :

After factorial theorem you have to know what the binomial coefficient is a number that represents the number of ways to choose k items from a set of n items, without regard to the order of the items.It is denoted by C(n,r) or

Binomial Theorem :

The binomial theorem is a fundamental tool in algebra and calculus, and it is used in many different areas of mathematics and science. For example, it can be used to expand polynomials, to calculate the powers of binomials, and to evaluate combinations and permutations.The binomial theorem is a powerful tool that has many different applications in mathematics and science. It is used to expand polynomials, to calculate the powers of binomials, and to evaluate combinations and permutations. It is also used in calculus to find derivatives and integrals, and in probability and statistics to calculate the probability of certain events occurring.

The binomial theorem states that for any non-negative integer n and any real numbers a and b, the expansion of the binomial (a+b)^n is given by the sum of the terms: