# Binomial Theorem

## Binomial Expressions

As the power increases the expansion of terms becomes very lengthy and tedious to calculate. It can be easily calculated with the help of the Binomial Theorem.

## What is Binomial Theorem?

The binomial theorem in mathematics is the process of expanding an expression that has been raised to any finite power. A binomial theorem is a powerful tool of expansion, which is widely used in Algebra, probability, etc.

### Binomial Expression

A binomial expression is an algebraic expression that contains two dissimilar terms such as a + b, a³ + b³, etc.

### Binomial Theorem Expansion

According to the theorem, we can expand the power (x + y)$^{n}$ into a sum involving terms of the form ax$^{b}$y$^{c}$, where the exponents b and c are nonnegative integers with b+c=n and the coefficient a of each term is a specific positive integer depending on n and b.

The theorem is given by the formula:

(x + y)$^{n}$ = $\sum_{k=0}^{n}$ ($_{k}^{n}$) x$^{n-k}$y$^{k}$ = $\sum_{k=0}^{n}$ ($_{k}^{n}$) x$^{k}$y$^{n-k}$

(x + y)$^{n}$ = $\sum_{k=0}^{n}$ (nk) x$^{n-k}$y$^{k}$ = $\sum_{k=0}^{n}$ (nk) x$^{k}$y$^{n-k}$

### Binomial Theorem Rules

The coefficients that appear in the binomial expansion are known as binomial coefficients. These are usually written ($_{k}^{n}$) or $^{n}C_{k}$. which means n choose k.

The coefficient of a term x$^{n-k}$y$^{k}$ in a binomial expansion can be calculated using the combination formula. The formula consists of factorials:

($_{k}^{n}$) = $\frac{n!}{k!(n-k)!}$

Important Points to Remember While Solving Binomial Expansion:

• The total number of terms in the expansion of (x + y)$^{n}$ is (n+1)

• The sum of exponents is always equal to n i.e (x + y) = n.

• nC$_{0}$, nC$_{1}$, nC$_{2}$, … .., nC$_{n}$ is called binomial coefficients and also represented by C$_{0}$, CC$_{1}$, CC$_{2}$ ….., C$_{n}$ respectively.

• The binomial coefficients which are equidistant from the beginning and from the ending are of equal value i.e. nC0= nCn,nC1= nCn-1 , nC2= nCn-2 ,….. etc.

• To find binomial coefficients we can also use Pascal’s Triangle. Some Other Useful Expansions that Help in an Easy Way to Solve Binomial Theorem :

• (x + y)$^{n}$ + (x - y)$^{n}$ = 2[C$_{0}$x$^{n}$ + C$_{2}$x$^{(n-1)}$y$^{2}$ + C$_{4}$x$^{n-4}$y$^{4}$+ …]

• (x + y)$^{n}$ - (x - y)$^{n}$ = 2[C$_{1}$ x$^{(n-1)}$y + C$_{3}$ x$^{(n-3)}$y$^{3}$ + C$_{5}$x$^{(n-5)}$y$^{5}$...]

• (1 + x)$^{n}$ = [C$_{0}$ + C$_{1}$x + C$_{2}$x$^{2}$ + … C$_{n}$x$_{n}$]

• (1 + x)$^{n}$ + (1 - x)$^{n}$ = 2[C$_{0}$ + C$_{2}$x$^{2}$ + C$_{4}$x$^{4}$ + …]

• (1 + x)$^{n}$ - (1 - x)$^{n}$ = 2[C$_{1}$x + C$_{3}$x$^{3}$ + C$_{5}$x$^{5}$ + …]

• The number of terms in the expansion of (x + a)$^{n}$ + (x - a)$^{n}$ is (n+2)/2 if “n” is even or (n+1)/2 if “n” is odd.

• The number of terms in the expansion of (x + a)$^{n}$ - (x - a)$^{n}$ is (n/2) if “n” is even or (n+1)/2 if “n” is odd.

Properties of Binomial Coefficients

Binomial coefficients refer to the integers that are coefficients in the binomial theorem. Some of the important properties of binomial coefficients are given below:

• C$_{0}$ + C$_{1}$ + C$_{2}$ + … + C$_{n}$ = 2n

• C$_{0}$ + C$_{2}$ + C$_{4}$ + … = C$_{1}$ + C$_{3}$ + C$_{5}$... = 2n - 1

• C$_{0}$ - C$_{1}$ + C$_{2}$ - C$_{3}$ + … + (-1)$^{n}$ . nC$_{n}$ = 0

• nC$_{1}$ + 2nC$_{2}$ + 3.nC$_{3}$ + … +n.nC$_{n}$ = n.2$^{(n-1)}$.

• C$_{1}$ - 2C$_{2}$ + 3C$_{3}$ - 4C$_{4}$+ … +(-1)$^{(n-1)}$C$_{n}$ = 0 for n ＞ 1

• C$_{0}$ $^{2}$ + C$_{1}$ $^{2}$ + C$_{2}$ $^{2}$ + ...C$_{n}$ $^{2}$ = $\frac{[(2n)!]}{(n!)^{2}}$

Terms in the Binomial Expansion

In binomial expansion, we generally find the middle term or the general term. The different Binomial Term involved in the binomial expansion is:

• General Term

• Middle Term

• Independent Term

• Determining a Particular Term

• Numerically greatest term

• The ratio of Consecutive Terms/Coefficients

General Term in Binomial Expansion:

We have (x + y)$^{n}$ = nC$_{0}$x$^{n}$ + nC$_{1}$x$^{(n-1)}$y + nC$_{2}$x$^{n-2}$y$^{2}$ + … + nC$_{n}$Y$^{n}$

General Term = T$_{(r+1)}$ = nC$_{r}$x$^{n-r}$ .y$^{r}$

General Term in (1+x)$^{n}$ nC$_{r}$x$^{r}$

In the binomial expansion of (x + y)$^{n}$, the r$^{th}$ term from the end is (n - r + 2)$^{th}$.

Middle Term(S) in the expansion of (x + y)$^{n,n}$

If n is even then (n/2 + 1) term is the middle term.

If n is odd then [(n+1)/2]$^{th}$ and [(n+3)/2)$^{th}$ terms are the middle terms of the expansion.

### Applications of Binomial Theorem

The binomial theorem has various applications in mathematics like finding the remainder, finding digits of a number, etc. The most common binomial theorem applications are:

• Finding Remainder using Binomial Theorem.

• Finding Digits of a Number.

• Relation Between two Numbers.

• Divisibility Test.

Binomial Theorem Problems are explained with the help of Binomial theorem formula examples which is given below:

1. Find the coefficient of x$^{9}$ in the expansion of (1 + x) (1 + x$^{2}$) (1 + x$^{3}$) . . . . . . (1 + x$^{100}$).

Sol: x$^{9}$ can be formed in 8 ways.

i.e., x$^{9}$ x$^{(1+8)}$ x$^{(2+7)}$ x$^{(3+6)}$ x$^{(4+5)}$, x$^{(1+3+5)}$, x$^{(2+3+4)}$

∴ The coefficient of x$^{9}$ = 1 + 1 + 1 + . . . . + 8 times = 8.

2.Find the number of terms in (1 + 2x + x$^{2}$)$^{50}$.

Sol: Use formula of the number of terms (1 + 2x + x$^{2}$)$^{50}$ = [(1 + x)$^{2}$]$^{50}$ = (1 + x)$^{100}$

Hence, the number of terms = (100 + 1) = 101

3. Find the value of $\binom{6}{2}$ using a pascal triangle.

Solution:

Look at the 2nd element in the 6th row in pascal's triangle. The value of $\binom{6}{2}$ will be that element. Hence, the value of $\binom{6}{2}$ is $15$.

4. Expand $( a + 2)^6$ using binomial theorem.

Solution:

Let $a = x, y = 2$ and $n = 6$

Substituting the values on binomial formula, we get

$(a)^6+6(a)^5(2)+\dfrac{6(5)}{2\text{!}}(a)^4(2)^{2}+\dfrac{6(5)(4)}{3\text{!}}(a)^3(2)^3+\dfrac{6(5)(4)(3)}{4\text{!}}(a)^2(2)^4+\dfrac{6(5)(4)(3)(2)}{5\text{!}}(a)(2)^5+ 2^6$

$= a^6 + 12a^5 + 60a^4 +160a^3 + 240a^2 + 192a + 64$

### Facts to Remember

• The binomial theorem was invented by Issac Newton.

• The Pascal triangle was invented by Blaise Pascal.

• The numbers in each row in the pascal triangle are known as the binomial coefficients.

• The numbers on the second diagonal and third diagonal in the pascal triangle form counting numbers and triangular numbers respectively.

• The sum of the numbers on each row are powers of $2$ whereas a series of diagonals of pascal's triangle forms the Fibonacci Sequence.