Binomial Theorem
Binomial theorem primarily helps to find the expanded value of the algebraic expression of the form (x + y)^{n}. Finding the value of (x + y)^{2}, (x + y)^{3}, (a + b + c)^{2} is easy and can be obtained by algebraically multiplying the number of times based on the exponent value. But finding the expanded form of (x + y)^{17} or other such expressions with higher exponential values is possible with the help of the binomial theorem. The exponent value of this binomial theorem expansion can be a negative number or a fraction. Here we limit our explanations to only nonnegative values. Let us learn more about the terms and the properties of coefficients in this binomial expansion.
What is Binomial Theorem?
The first mention of the binomial theorem was in the 4th century BC by a famous Greek mathematician by name of Euclids. The binomial theorem states the principle for expanding the algebraic expression (x + y)^{n} and expresses it as a sum of the terms involving individual exponents of variables x and y. Each term in a binomial expansion is associated with a numeric value which is called coefficient.
Statement: According to the binomial theorem, it is possible to expand any nonnegative power of binomial (x + y) into a sum of the form,
\((x+y)^{n} = {n \choose 0}x^{n}y^{0}+{n \choose 1}x^{n1}y^{1}+{n \choose 2}x^{n2}y^{2}+\cdots +{n \choose n1}x^{1}y^{n1}+{n \choose n}x^{0}y^{n}\)
where, \(n\geq 0\) is an integer and each \( {\tbinom {n}{k}}\) is a positive integer known as a binomial coefficient.
Note: When an exponent is zero, the corresponding power expression is 1. This multiplicative factor is often omitted from the term, therefore often the right hand side is directly written as \({\binom {n}{0}}x^{n}+\ldots\). This formula is also referred to as the binomial formula or the binomial identity. Using summation notation, the binomial theorem can be given as,
\((x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{nk}y^{k}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{nk}\)
Example: Let us expand (x+3)^{5} using the binomial theorem. Here a = 3 and n = 5. Substituting and expanding, we get:
(x+3)^{5 }= \(^5C_{0} x^{50}3^0 + ^5C_{1 }x^{51}3^1 + ^5C_{2} x^{52}3^2 + ^5C_{3} x^{53}3^3 +^5C_{4} x^{54}3^4 + ^5C_{5} x^{55}3^5\)
= x^{5}^{ }+ 5_{ }x^{4}. 3^{ }+ 10_{ }x^{3 }. 9 + 10_{ }x^{2} . 27^{ }+ 5x .81 +^{ }_{ }3^{5}
= x^{5}^{ }+ 15_{ }x^{4 }+ 90x^{3} + 270_{ }x^{2 }+ ^{ }405 x + 243
Binomial Expansion
Binomial expansion of (x + y)^{n }by using the binomial theorem is as follows,
(x + y)^{n} = \(^nC_0x^n + ^nC_1x^{n  1}y + ^nC_2x^{n  2}y^2 + .....+^nC_rx^{n  r}y^r + ......^nC_ny^n\).
Binomial Theorem Formula
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is \((a+b)^{n}=\sum_{r=0}^{n}(^nC_{r})a^{nr}b^{r}\), where n is a positive integer and a, b are real numbers and 0 < r ≤ n. This formula helps to expand the binomial expressions such as x + a, (2x + 5)^{3}, (x^{ } (1/x))^{4}, and so on.^{ }
The binomial theorem formula helps in the expansion of a binomial raised to a certain power. Let us understand the binomial theorem formula and its application in the following sections.
The binomial theorem states: If x and y are real numbers, then for all n ∈ N,
(x + y)^{n} = \(^nC_0x^n + ^nC_1x^{n  1}y + ^nC_2x^{n  2}y^2 + .....+^nC_rx^{n  r}y^r + ......^nC_nx^0y^n\)
⇒ \((x+ay)^{n}=\sum_{r=0}^{n}(^nC_{r})x^{nr}y^{r} \)
where, \(^nC_{r}= \dfrac{n!}{ r ! (nr)!}\)
Binomial Theorem Proof
Let x, a, n ∈ N. Let us prove the binomial theorem formula through induction. It is enough to prove for n = 1, n = 2, for n = k ≥ 2, and for n = k+ 1.
It is obvious that (x +y)^{1} = x +y and
(x +y)^{2} = (x + y) (x +y)
= x^{2} + xy +xy + y^{2} (using distributive property)
x^{2} +2xy + y^{2}
Thus the result is true for n = 1 and n = 2. Let k be a positive integer. Let us prove the result is true for k ≥ 2.
Assuming \((x+y)^{n}=\sum_{r=0}^{n}(^nC_{r})x^{nr}y^{r} \),
\((x+y)^{k}=\sum_{r=0}^{k}(^kC_{r})x^{kr}y^{r} \),
\((x+y)^{k}= ^nC_0x^k + ^kC_1x^{k  1}y + ^kC_2x^{k  2}y^2 + .....+^kC_rx^{k  r}y^r + ......^kC_ky^k\).
\((x+y)^{k}= x^k + ^kC_1x^{k  1}y + ^kC_2x^{k  2}y^2 + .....+^kC_rx^{k  r}y^r + .....y^k\)
Thus the result is true for n = k ≥ 2.
Now consider the expansion for k + 1.
(x + y)^{ k+1 }= (x + y) (x + y)^{k}
\(= (x +y)(x^k + ^kC_1x^{k  1}y + ^kC_2x^{k  2}y^2 + .....+^kC_rx^{k  r}y^r + ......y^k)\)
\(= x^{k+1} + [1 + ^kC_{1}] x^k y + [^kC_{1 }+^kC_{2}] x ^{k1} y^2 +........+\\\\ [ ^kC_{r1} + ^kC_{r}] x ^{kr+1} y ^r + .........[^k C _{k1 }+ 1] xy^k +y^{k+1}\)
\(= x^{k+1 }+ ^{k+1} C _{1}x^{k} y +...... ^{k+1}C_{r} x ^{k+1r} y^r + ....... ^{k+ 1}C _{k}xy^k +y^{k+1}\) [\(\because ^nC_r + ^nC_{r1} = ^{n+1}C_r\)]
Thus the result is true for k+1.
By mathematical induction, this result is true for all positive integers 'n'. Hence the proof.
Properties of Binomial Theorem
 The number of coefficients in the binomial expansion of (x + y)^{n} is equal to (n + 1).
 There are (n+1) terms in the expansion of (x+y)^{n}.
 The first and the last terms are x^{n }and y^{n} respectively.
 From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
 The general term in the expansion of (x + y) ^{n }is the (r +1)^{th} term that can be represented as \(T_{r+1}\), \(T_{r + 1} = ^nC_rx^{n  r}y^r\)
 The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.
 In the binomial expansion of (x + y)^{n}, the r^{th} term from the end is (n – r + 2)^{th} term from the beginning.
 If n is even, then in (x + y)^{n} the middle term = (n/2)+1 and if n is odd, then in (x + y)^{n}, the middle terms are (n+1)/2 and (n+3)/2.
Binomial Theorem Coefficients
The binomial coefficients are the numbers associated with the variables x, y, in the expansion of (x + y)^{n}. The binomial coefficients are represented as \(^nC_0\), \(^nC_1\), \(^nC_2\) ..... The binomial coefficients are obtained through the pascal triangle or by using the combinations formula.
Pascal's Triangle
The values of the binomial coefficients exhibit a specific trend which can be observed in the form of Pascal's triangle. Pascal's triangle is an arrangement of binomial coefficients in triangular form. It is named after the French mathematician Blaise Pascal. The numbers in Pascal's triangle have all the border elements as 1 and the remaining numbers within the triangle are placed in such a way that each number is the sum of two numbers just above the number.
Combinations
The formula for combinations is used to find the value of the binomial coefficients in the expansions using the binomial theorem. The combinations in this case are the different ways of picking r variables from the available n variables. The formula to find the combinations of r objects taken from n different objects is \(^nC_r = \frac{n!}{r!(n  r)!}\). Here the coefficients have the following properties.
 \(^nC_0 = ^nC_n = 0\)
 \(^nC_1 = ^nC_{n  1} = n\)
 \(^nC_r = ^nC_{r  1}\)
The following properties of binomial expansion can be derived by simply substituting simple numeric values of x = 1 and y = 1 in the binomial expansion of (x + y)^{n}. The properties of binomial coefficients are as follows.
 \(C_1 + C_2 + C_3 + C_4 + .......C_n = 2^n\)
 \(C_0 + C_2 + C_4 + .... = C_1 + C_3 + C_5 + ....... = 2^{n  1}\)
 \(C_0  C_1 + C_2  C_3 + C_4  C_5 + ..........(1)^nC_n = 0\)
 \(C_1 + 2C_2 + 3C_3 + 4C_4.......... + nC_n = n2^{n  1}\)
 \(C_1  2C_2 + 3C_3  4C_4 .........+ (1)^{n  1} nC_n = 0\)
 \(C_0^2 + C_1^2 + C_2^2 + C_3^2 + C_4^2 + ......C_n^2 = \dfrac{(2n)!}{(n!)^2}\)
Important Terms of Binomial Expansion
The following terms related to binomial expansion using binomial theorem are helpful to find the terms. The details of each of the terms are as follows.
General Term: This term symbolizes all of the terms in the binomial expansion of (x + y)^{n}. The general term in the binomial expansion of (x + y)^{n} is \(T_{r + 1} = ^nC_rx^{n  r}y^r\). Here the rvalue is one less than the number of the term of the binomial expansion. Also, \(^nC_r\) is the coefficient, and the sum of the exponents of the variables x and y is equal to n.
Middle Term: The total number of terms in the expansion of (x + y)^{n}^{ }is equal to n + 1. The middle term in the binomial expansion depends on the value of n. The middle term and the number of middle terms depend on the value of n: n is even or odd. For an even value of n there is only one middle term and (n/2 + 1)^{th} term is the middle term. For an odd value of n, there are two middle terms, and the two middle terms are n/2, and n/2 + 1 terms.
Identifying a Particular Term: There are two simple steps to identify a particular term containing x^{p}. First, we need to find the general term in the expansion of (x + y)^{n}. which is is \(T_{r + 1} = ^nC_rx^{n  r}y^r\). Secondly, we need to compare this with x^{p} to obtain the rvalue. Here the rvalue is helpful to find the particular term in the binomial expansion. Let us find the fifth term in the expansion of (2x + 3)^{9} using the binomial theorem.
The formula to find the n^{th} term in the binomial expansion of (x + y)^{n} is \(T_{r + 1} = ^nC_rx^{n  r}y^r\).
Applying this to (2x + 3)^{9} , \(T_5 = T_{4 + 1} = ^9C_4(2x)^{9  4}.3^4\). Thus the 5th term is = \(^9C_4(2x)^5.3^4\)
Term Independent of X: The steps to find the term independent of x is similar to finding a particular term in the binomial expansion. First, we need to find the general term in the expansion of (x + y)^{n}. which is \(T_{r + 1} = ^nC_rx^{n  r}y^r\). Here to find the term independent of x, we need to find and equalize the exponent of x in the general term to zero. From the obtained value of r, the term independent of x is (r + 1)^{th} term.
Numerically Greatest Term: The formula to find the numerically greatest term in the expansion of (1 + x)^{n} is \(\dfrac{(n + 1)x}{1 + x}\). There are two points to be remembered while using this formula to find the numerically greatest term. First, we need to convert any binomial expansion into the form of (1 + x)^{n}. We can convert (2x + 3y)^{5} to (1 + 3y/2x)^{5} . Further for this expansion x is the numeric value and is equal to 3/2 in this given example. The final answer is rounded to the integral value to obtain the numerically greatest term.
Binomial Expansion for Negative Exponent
The binomial theorem expansion also applies to exponents with negative values. The standard coefficient values of binomial expansion for positive exponents are the same for the expansion with the negative exponents. The terms and the coefficient values remain the same, but the algebraic relationship between the terms varies in the binomial expansion of negative exponents.
 (1 + x)^{1} = 1  x + x^{2}  x^{3} + x^{4}  x^{5} + .......
 (1  x)^{1} = 1 + x + x^{2} + x^{3} + x^{4} + x^{5} + .......
 (1 + x)^{2} = 1  2x + 3x^{2}  4x^{3} + ........
 (1  x)^{2} = 1 + 2x + 3x^{2} + 4x^{3} + ........
 (1 + x)^{3} = 1  3x + 6x^{2}  10x^{3} + 15x^{4} + ......
 (1  x)^{3} = 1 + 3x + 6x^{2} + 10x^{3} + 15x^{4} + ......
☛ Also Check:
 Algebraic Identities
 Binomial Distribution Formula
 Binomial Distribution
 Probability and Statistics
 Geometric Distribution Formula
Important Notes
The following key points would help in a better understanding of the binomial theorem.
 The number of terms in the binomial expansion of (x + y)^{n} is equal to n + 1.
 In the expansion of (x + y)^{n}, the sum of the powers of x and y in each term is equal to n.
 The value of the binomial coefficients from either side of the expansion is equal.
 The number of terms in the binomial expansion of (x + y + z)^{n} is n(n + 1).
Binomial Theorem Examples

Example 1: What is the binomial expansion of (x^{2} + 1)^{5} using the binomial theorem?
Solution:
The following formula derived from the Binomial Theorem is helpful to find the expansion.
(x + y)^{n} = \(^nC_0x^n + ^nC_1x^{n  1}y + ^nC_2x^{n  2}y^2 + .....+^nC_rx^{n  r}y^r + ......^nC_ny^n\).
(x^{2} + 1)^{5} = \(^5C_0(x^2)^5 +^5C_1(x^2)^4 + ^5C_2(x^2)^3 + ^5C_3(x^2)^2 + ^5C_4(x^2)^1 + ^5C_5 \)
= \(1.x^{10} +5.x^8 + 10.x^6 + 10.x^4 +5.x^2 + 1 \)
= \(x^{10} +5.x^8 + 10.x^6 + 10.x^4 +5.x^2 + 1 \)
Answer: (x^{2} + 1)^{5} = \(x^{10} +5.x^8 + 10.x^6 + 10.x^4 +5.x^2 + 1 \).

Example 2: Find the 7^{th} term in the expansion of (x + 2)^{10}
Solution:
The general term in the expansion of (x+y)^{n }using the binomial theorem formula is
\(T_{r + 1} = ^nC_rx^{n  r}y^r\).
Here r = 6, n =10, a = 2
Thus by substituting, we get
\(T7 = T_{6 + 1} = ^{10}C_6x^{106}2^6\)
T_{7 }= 210 x ^{4 }. 64
= 13440 x^{4}
Answer:13440 x^{4}

Example 3: Find the coefficient of x^{2 }in (x +(1/x))^{8}
Solution:
Using the binomial theorem formula in the expansion of (x +1/x)^{8}, we have x^{2 }as the fourth term.
\(^8C_{0} x^8 (1/x)^0 + ^8C_{1} x^7 (1/x)^1 + ^8C_{2} x^6 (1/x)^2 + ^8C_{3} x^5 (1/x)^3\\\\ + ^8C_{4} x^4 (1/x)^4+ ^8C_{5} x^3(1/x)^5 + ^8C_{6} x^2 (1/x)^6 + ^8C_{7} x^1 (1/x)^7 + ^8C_{8} x^0 (1/x)^8\)
The coefficient of the fourth term is \(^8C_{3}\)= 56
Answer: The coefficient of x^{2} in the expansion of (1+ (1/x))^{8}= 56
FAQs on Binomial Theorem
What Is the Binomial Theorem?
The binomial theorem is used for the expansion of the algebraic terms of the form(x + y)^{n} . (x + y)^{n} = \(^nC_0x^n + ^nC_1x^{n  1}y + ^nC_2x^{n  2}y^2 + .....+^nC_rx^{n  r}y^r + ......^nC_ny^n\). Here the number of terms in the binomial expansion having an exponent of n is n + 1. The exponent of the first term in the expansion is decreasing and the exponent of the second term in the expansion is increasing in a progressive manner. The coefficients of the binomial expansion can be found from the pascals triangle or using the combinations formula of \(^nC_r = \dfrac{n!}{r!.(n  r)!}\).
What Is the Constant Term in the Binomial Theorem?
The constant term in the binomial expansion is a numeric value and is independent of the variables. For a binomial expansion of (x + y)^{n} the term independent of x can be calculated by finding the term independent of x.
What Is the Coefficient in the Binomial Theorem?
The coefficients in the binomial expansion are \(^nC_0\), \(^nC_1\), \(^nC_2\), .....\(^nC_r\) ......\(^nC_n\). The coefficient value for n = 4 is equal to 1, 4, 6, 4,1, and the coefficient value for n = 5 is 1, 5, 10, 20, 10, 5, 1. The coefficient values can be found either from the pascals triangle or by using the combinations formula of \(^nC_r = \dfrac{n!}{r!.(n  r)!}\).
Where Is Binomial Theorem Used?
The binomial theorem is useful to do the binomial expansion and find the expansions for the algebraic identities. Further, the binomial theorem is also used in probability for binomial expansion. A few of the algebraic identities derived using binomial theorem is as follows.
 (a + b)^{2} = a^{2} + 2ab + b^{2}
 (a  b)^{2} = a^{2}  2ab + b^{2}
 (a + b)(a  b) = a^{2}  b^{2}
 (a + b)^{3} = a^{3} +3a^{2}b + 3ab^{2} + b^{3}
 (a  b)^{3} = a^{3}  3a^{2}b + 3ab^{2}  b^{3}
 (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ac
What Is the General Term in Binomial Theorem?
The general term of the binomial expansion of (x + y)^{n} is \(T_{r + 1} = ^nC_rx^{n  r}y^r\). The rvalue for the term is one less than the number of the term. In the general term the sum of the exponents of both the terms is equal to n. Further, the general term is helpful to derive the independent term, numerically greatest term, and the term with particular power.
What Is the Middle Term in Binomial Theorem?
The middle term in the binomial expansion depends on the value of n. The middle term and the number of middle terms depend on the value of n: n is even or odd. For an even value of n there is only one middle term and (n/2 + 1) term is the middle term. For an odd value of n, there are two middle terms and the two middle terms are n/2, and n/2 + 1 terms.
How to Find Terms in Binomial Theorem?
The number of terms in a binomial expansion with an exponent of n is equal to n + 1. Further to find a particular term in the expansion of (x + y)^{n} we make use of the general term formula. The general term of the binomial expansion is \(T_{r + 1} = ^nC_rx^{n  r}y^r\) . Here the coefficient values are found from the pascals triangle or using the combinations formula, and the sum of the exponents of both the terms in the general term is equal to n.
How to Find Number of Terms in Binomial Theorem?
The number of terms in the binomial expansion is one more than the exponent value. The number of terms in the expansion of (x + y)^{n} is equal to (n + 1).
What Is the General Form of Binomial Theorem Formula?
the general formula of binomial theorem formula is used in the expansion of binomial expression of the form:
(x + a)^{n} = \(^nC_0x^n + ^nC_1x^{n  1}a + ^nC_2x^{n  2}a^2 + .....+^nC_rx^{n  r}a^r + ......^nC_na^n\),
where,
 x, a, n are natural numbers, and,
 0 < r ≤ n
Where Is Binomial Theorem Formula Used?
The binomial theorem formula is used to expand the binomial expressions. We apply the formula in finding probability, combinatorics, calculus, and in other important areas of math. For example, (101)^{5 }=(100+1)^{5 }= 100^{5}+ 5 × 100^{4 }+ 10 × 100^{3}+ 10 × 100^{2}+ 5 × 100 + 1 = 10,000,000,000+ 500,000,000 + 10,000,000 + 100,000 + 500 +1 = 10,510,100,501
What Is n and r in The Binomial Theorem Formula?
In the binomial theorem formula of expansion (x+a)^{n}, we use the combinatorics formula that is denoted as \(^nC_{r}\)_{, }where n is the exponent in the expansion_{ }and_{ }r is the term number that ranges from 0 to n.
How Do You Find the Number of Terms in the Expansion using the Binomial Theorem Formula?
The number of terms in the binomial expansion of (x+a)^{n }is n+1 terms. i.e. add 1 to the power, the binomial coefficient is raised to. If we expand (2x+5)^{10}, we will have 10 +1 = 11 terms on expansion.