There are various types of theorems in mathematics such as the Pythagoras theorem, the caves theorem, the Bayes theorem, and many more. Similarly, the binomial theorem is a type of theorem used to find the value of an algebraic expression in the form of ( a + b )n where ‘a’ and ‘b’ are the variables of the equation. You might find ‘x’ and ‘y’ in various other parts, but do not get confused. The binomial theorem produces an exponent value which can always be a fraction of a number that is negative in nature. The use of this theorem is to find expanded values of some expression consisting of higher exponential values. The expressions with lower exponential values can be easily obtained algebraically. In this article, we will try to cover some basic concepts regarding binomial theorem such as a binomial theorem in pascals triangle, points to remember and a detailed analysis.

#### Binomial Theorem in Pascal’s Triangle

The arrangement of binomial coefficients in a form of triangle or triangular form can be defined as the pascal’s triangle. The term was named after the name of a great mathematician and physicist, ‘Blaise Pascal’. The use of Pascal’s triangle ranges from probability theory to algebra and combinatorics. In this paragraph, we may know how the binomial theorem is related to Pascal’s triangle.

Whenever you observe Pascal’s triangle, you will see that the coefficient of the binomial theorem is exhibiting a unique trend which will be observed in the triangle. There are some special patterns seen in the triangle such as the addition of the diagonals row will give use the resultant value which may form the Fibonacci series, if any row consists of a prime number of the second element, then all the elements present in that particular row will be divisible by the prime number mentioned, every row in the triangle have special numbers in them like, natural numbers, triangular numbers, tetrahedral numbers and many more.

#### Some Significant Points to Remember about Binomial Theorem

As mentioned above, the binomial theorem is a type of theorem used to find the value of an algebraic expression in the form of ( a + b )n where ‘a’ and ‘b’ are the variables of the equation. In this section, we will try to sum up every point about the binomial theorem that we have discussed.

- The exponent value present in the binomial theorem always produces a fraction or a negative number.
- The binomial theorem was invented by the famous mathematician, ‘Euclids’ around the 4th Century BC (before christ ).
- The binomial coefficient is arranged in the form of a triangle also known as Pascal’s triangle. This triangle is named after the great French mathematician, Blaise Pascal.
- The formula stated for combinations can be used to calculate the values of coefficients in expansion which can be formulated using the binomial theorem.
- The resultant value of the coefficients ( binomial ) from the one side in expansion is always equal. The quantity of terms in the expansion is equivalent to n(n+1).

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