Properties of Binomial Coefficients

Mathematics covers the vast syllabus for the preparation of the IIT-JEE. The topic of binomial theorems and properties of binomial coefficients is vast and contains many rules and formulas. This topic may not be very simple, but understanding it step by step will be a good investment of time for the exams.

Definition:

The binomial theorem is a method to derive an expanded value of any algebraic expression. To understand the properties of binomial coefficients, it is essential to understand the basics of the binomial theorem. The derivation of the binomial theorem is also the parallel topic for this.

This expression can have a face value like (x + y)n. The value of n in such an expression is the exponential value, an independent term of the algebraic expression. The value of the exponent in any of the expressions can also be with a negative value and in fraction form.

The binomial theorem helps to find the higher values of the exponents by their variable properties of binomial coefficients. Generally, the exponential values in any expression have a positive charge but sometimes, and it may be with the negative value.

Basic terms for binomial expansion:

The binomial theorem and its calculations are to get the exponential value of an expression. The algebraic form of expression sometimes demands the special terminology of an exponential value. They have the categorisation for the value of the exponents according to the properties of binomial coefficients. Here are the listings:

• General term
• Middle term
• Independent term
• Determination of particular term
• Greatest numerical term
• The ratio for consecutive term

These terms have different values and methods to get the exponential values following the properties of binomial coefficients. The explanation of these terms needs a certain expression in which we can determine the type of term and its binomial coefficient. Here is the explanation:

• General term:

The term has the definition according to the variable expression. Here in this expression: 

(x + y)n = nC0 xn.y0 + nC1 xn-1.y + nC2 xn-2.y2 + … + nCn x0.yn the derivation of binomial coefficient and the general term is distinct. The general term here is Tr+1 = nCr xn-r.yr.

This format of the term is to find the number of terms in an algebraic expression. In some cases, the term’s position and its value are also visible.

• Middle term:

In the binomial expansions, determining the middle term depends on the even and odd value of terms. While describing with a mathematical expression of (x+y)n, the value of the term differs when the value of n is even the middle term comes as the (n/2 + 1)th term. But when the value of n is odd, it has the variable middle term. These variable values are [(n+1)/2]th and [(n+3)/2)th.

Independent term:

This method is to find the integer or independent term in the algebraic expression. The only basic rule is that the value of the integer will always be positive. The numerical explanation of this term is as in this, [axp + (b/xq)]n the independent term is Tr+1 = nCr an-r br, where the integer is r = (np/p+q), which have the positive value.

Determination of particular term:

In this type of binomial expansion, the calculation is to find the value of each term in a particular and separate manner. The mathematical [axp + (b/xq)]n here the coefficient value of xm is equal to the coefficient value of Tr+1. here the value of r= [(np−m)/(p+q)].

Greatest numerical term:

To find the binomial coefficient and the numerically greatest term depends on the type of integer present in the algebraic expression. In this solution, the calculation is to find the greatest term in the expression of the integers present in series. The numerical representation for the binomial expansion of (1+x)n has this term as [ (n+1) |x| ] / [ |x| + 1]  = P. here, P is the positive integer of the expression. The value of the term must be greater than zero during the expansion for the properties of Binomial coefficients.

The ratio of consecutive terms:

Here in this type of binomial expansion with the properties of binomial coefficients, the calculation is to find the constant ratio between the two consecutive coefficients of the binomial expression.

Properties of Binomial coefficients:

Here are some of the properties of Binomial coefficients which are necessary to keep in mind while calculating the binomial coefficients. Here are the mentions:

• The number of terms or coefficients in the (x + y)n for the binomial expansion will be equal to (n+1).

• This is one of the properties of binomial coefficients in which the first term of the binomial expansion is always xn, and the last term will always be yn.

• The arrangement of the binomial terms will be according to the Pascal triangle, which follows the array arrangement method and develops the pattern for the binomial theorem.

• The expansion has (n+1) elements that are one greater than this index.

Conclusion:

The binomial theorem and the Properties of Binomial coefficients is a complex topic with various distinctive methods of solutions. But on the other hand, this is also a very important topic in the syllabus of IIT-JEE. It covers most of the syllabus and impacts strongly on the results.