The Binomial Theorem
by aoum, Mar 22, 2025, 10:29 PM
The Binomial Theorem: Expanding Powers of Binomials
The Binomial Theorem is a fundamental theorem in algebra and combinatorics that provides a systematic way to expand powers of binomials. It connects algebraic expressions to combinatorial ideas and is a cornerstone of mathematical analysis. This article explores the statement, proof, properties, and applications of the Binomial Theorem in detail.
1. Statement of the Binomial Theorem
For any nonnegative integer
and any real or complex numbers 
and 
, the th power of the binomial 
can be expanded as:
![\[
(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k
\]](//latex.artofproblemsolving.com/6/b/a/6ba58694a4b7afcd41684dd4880842c82c6f0311.png)
Where:
The binomial coefficient is defined by the formula:
![\[
\binom{n}{k} = \frac{n!}{k! (n-k)!}
\]](//latex.artofproblemsolving.com/4/2/a/42a4b80aca41d6a3bd8357918559e53c089cffc0.png)
2. Understanding the Binomial Expansion
The Binomial Theorem tells us that the expansion of
consists of 
terms. Each term has:
For example:
![\[
(x + y)^4 = \sum_{k=0}^{4} \binom{4}{k} x^{4-k} y^k
\]](//latex.artofproblemsolving.com/4/9/1/491c8312b87a69f5fc98cd6a0923b9f04a10eacb.png)
Expanding the sum:
![\[
(x + y)^4 = \binom{4}{0} x^4 + \binom{4}{1} x^3 y + \binom{4}{2} x^2 y^2 + \binom{4}{3} x y^3 + \binom{4}{4} y^4
\]](//latex.artofproblemsolving.com/2/e/1/2e1fe1aa1e55fab5dbc31920ddaebbb08fda1c7e.png)
Using the values of the binomial coefficients:
![\[
(x + y)^4 = 1x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + 1y^4
\]](//latex.artofproblemsolving.com/b/1/a/b1a47fd923f1ec1bff03e81c2963a0a147edd519.png)
3. Proof of the Binomial Theorem
We prove the Binomial Theorem using mathematical induction.
Base Case: For
,
![\[
(x + y)^0 = 1
\]](//latex.artofproblemsolving.com/a/4/2/a42e95e712fa2ab34d57e21cd2d54f80e8690833.png)
And:
![\[
\sum_{k=0}^{0} \binom{0}{k} x^{0 - k} y^k = \binom{0}{0} x^0 y^0 = 1
\]](//latex.artofproblemsolving.com/4/7/2/47206a7506d64b5da9218ef03ee371ff5e88ef46.png)
The base case holds.
Inductive Step: Assume the Binomial Theorem is true for some
, i.e.,
![\[
(x + y)^m = \sum_{k=0}^{m} \binom{m}{k} x^{m - k} y^k
\]](//latex.artofproblemsolving.com/8/5/2/85266abe3fbff33b88395cd65fa2f06b9ffbd45b.png)
We need to prove it for
:
![\[
(x + y)^{m + 1} = (x + y) \cdot (x + y)^m
\]](//latex.artofproblemsolving.com/9/4/8/948cec03e96802a7daf5d11777ae9c934599eb4c.png)
By the induction hypothesis:
![\[
(x + y)^{m + 1} = (x + y) \sum_{k=0}^{m} \binom{m}{k} x^{m - k} y^k
\]](//latex.artofproblemsolving.com/e/b/2/eb2bbe7a449b167ce244f00c70b34c76bcecbd09.png)
Distribute:
![\[
= \sum_{k=0}^{m} \binom{m}{k} x^{m - k + 1} y^k + \sum_{k=0}^{m} \binom{m}{k} x^{m - k} y^{k + 1}
\]](//latex.artofproblemsolving.com/9/5/a/95abd4c3f04a2fa3c9a8394a82fdb46419bddfc7.png)
Reindexing the second sum:
![\[
= \sum_{k=0}^{m} \binom{m}{k} x^{m - k + 1} y^k + \sum_{k=1}^{m + 1} \binom{m}{k - 1} x^{m - (k - 1)} y^k
\]](//latex.artofproblemsolving.com/0/4/b/04b4336e4225fd56067ea791be796765f75500d2.png)
Using Pascal's Identity:
![\[
\binom{m}{k} + \binom{m}{k - 1} = \binom{m + 1}{k}
\]](//latex.artofproblemsolving.com/e/c/2/ec20977365c3a92df68e9dabde94477260abd433.png)
Thus:
![\[
(x + y)^{m + 1} = \sum_{k=0}^{m + 1} \binom{m + 1}{k} x^{(m + 1) - k} y^k
\]](//latex.artofproblemsolving.com/1/2/f/12f6fc2452c70d647c9d8725ab2094d7592065bb.png)
By induction, the Binomial Theorem holds for all nonnegative integers.
4. Properties of Binomial Coefficients
The binomial coefficients have several useful properties:
5. Generalized Binomial Theorem
When is any real or complex number, the Binomial Theorem can be extended to the form of an infinite series:
![\[
(1 + x)^\alpha = \sum_{k=0}^{\infty} \binom{\alpha}{k} x^k, \quad |x| < 1
\]](//latex.artofproblemsolving.com/f/a/e/faeec2c23ce7ae02e584d322b4f5c325b7c9bdb0.png)
Where:
![\[
\binom{\alpha}{k} = \frac{\alpha (\alpha - 1) (\alpha - 2) \dots (\alpha - k + 1)}{k!}
\]](//latex.artofproblemsolving.com/6/a/d/6ad28cdf508e9f6ef28e08240af15efc778a810e.png)
6. Applications of the Binomial Theorem
The Binomial Theorem has numerous applications:
7. Example Problems
8. Conclusion
The Binomial Theorem is a powerful and elegant mathematical tool that connects algebra, combinatorics, and analysis. Its applications span various fields, making it one of the most important results in elementary and advanced mathematics.
9. References
The Binomial Theorem is a fundamental theorem in algebra and combinatorics that provides a systematic way to expand powers of binomials. It connects algebraic expressions to combinatorial ideas and is a cornerstone of mathematical analysis. This article explores the statement, proof, properties, and applications of the Binomial Theorem in detail.
Visualization of binomial expansion up to the 4th power
1. Statement of the Binomial Theorem
For any nonnegative integer
and any real or complex numbers 
and 
, the th power of the binomial 
can be expanded as:![\[
(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k
\]](http://latex.artofproblemsolving.com/6/b/a/6ba58694a4b7afcd41684dd4880842c82c6f0311.png)
Where:
is the binomial coefficient, which represents the number of ways to choose 
objects from without regard to order.
represents the individual terms in the expansion.- The sum is taken over all integers from
to .
is the binomial coefficient, which represents the number of ways to choose 
objects from 
represents the individual terms in the expansion.
to The binomial coefficient is defined by the formula:
![\[
\binom{n}{k} = \frac{n!}{k! (n-k)!}
\]](http://latex.artofproblemsolving.com/4/2/a/42a4b80aca41d6a3bd8357918559e53c089cffc0.png)
2. Understanding the Binomial Expansion
The Binomial Theorem tells us that the expansion of
consists of 
terms. Each term has:- A binomial coefficient , which gives the number of ways to choose factors of from total factors.
- A power of , specifically
. - A power of , specifically
.
.
.For example:
![\[
(x + y)^4 = \sum_{k=0}^{4} \binom{4}{k} x^{4-k} y^k
\]](http://latex.artofproblemsolving.com/4/9/1/491c8312b87a69f5fc98cd6a0923b9f04a10eacb.png)
Expanding the sum:
![\[
(x + y)^4 = \binom{4}{0} x^4 + \binom{4}{1} x^3 y + \binom{4}{2} x^2 y^2 + \binom{4}{3} x y^3 + \binom{4}{4} y^4
\]](http://latex.artofproblemsolving.com/2/e/1/2e1fe1aa1e55fab5dbc31920ddaebbb08fda1c7e.png)
Using the values of the binomial coefficients:
![\[
(x + y)^4 = 1x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + 1y^4
\]](http://latex.artofproblemsolving.com/b/1/a/b1a47fd923f1ec1bff03e81c2963a0a147edd519.png)
3. Proof of the Binomial Theorem
We prove the Binomial Theorem using mathematical induction.
Base Case: For
,![\[
(x + y)^0 = 1
\]](http://latex.artofproblemsolving.com/a/4/2/a42e95e712fa2ab34d57e21cd2d54f80e8690833.png)
And:
![\[
\sum_{k=0}^{0} \binom{0}{k} x^{0 - k} y^k = \binom{0}{0} x^0 y^0 = 1
\]](http://latex.artofproblemsolving.com/4/7/2/47206a7506d64b5da9218ef03ee371ff5e88ef46.png)
The base case holds.
Inductive Step: Assume the Binomial Theorem is true for some
, i.e.,![\[
(x + y)^m = \sum_{k=0}^{m} \binom{m}{k} x^{m - k} y^k
\]](http://latex.artofproblemsolving.com/8/5/2/85266abe3fbff33b88395cd65fa2f06b9ffbd45b.png)
We need to prove it for
:![\[
(x + y)^{m + 1} = (x + y) \cdot (x + y)^m
\]](http://latex.artofproblemsolving.com/9/4/8/948cec03e96802a7daf5d11777ae9c934599eb4c.png)
By the induction hypothesis:
![\[
(x + y)^{m + 1} = (x + y) \sum_{k=0}^{m} \binom{m}{k} x^{m - k} y^k
\]](http://latex.artofproblemsolving.com/e/b/2/eb2bbe7a449b167ce244f00c70b34c76bcecbd09.png)
Distribute:
![\[
= \sum_{k=0}^{m} \binom{m}{k} x^{m - k + 1} y^k + \sum_{k=0}^{m} \binom{m}{k} x^{m - k} y^{k + 1}
\]](http://latex.artofproblemsolving.com/9/5/a/95abd4c3f04a2fa3c9a8394a82fdb46419bddfc7.png)
Reindexing the second sum:
![\[
= \sum_{k=0}^{m} \binom{m}{k} x^{m - k + 1} y^k + \sum_{k=1}^{m + 1} \binom{m}{k - 1} x^{m - (k - 1)} y^k
\]](http://latex.artofproblemsolving.com/0/4/b/04b4336e4225fd56067ea791be796765f75500d2.png)
Using Pascal's Identity:
![\[
\binom{m}{k} + \binom{m}{k - 1} = \binom{m + 1}{k}
\]](http://latex.artofproblemsolving.com/e/c/2/ec20977365c3a92df68e9dabde94477260abd433.png)
Thus:
![\[
(x + y)^{m + 1} = \sum_{k=0}^{m + 1} \binom{m + 1}{k} x^{(m + 1) - k} y^k
\]](http://latex.artofproblemsolving.com/1/2/f/12f6fc2452c70d647c9d8725ab2094d7592065bb.png)
By induction, the Binomial Theorem holds for all nonnegative integers
.4. Properties of Binomial Coefficients
The binomial coefficients have several useful properties:
- Symmetry:
![\[
\binom{n}{k} = \binom{n}{n - k}
\]](//latex.artofproblemsolving.com/b/4/2/b42a06fe41d2bdf0fb4d3fbec8ac0c702dfa896a.png)
- Sum of Binomial Coefficients:
![\[
\sum_{k=0}^{n} \binom{n}{k} = 2^n
\]](//latex.artofproblemsolving.com/f/1/b/f1b4bf6764470e55f4258a215a620efa9680917f.png)
- Alternating Sum:
![\[
\sum_{k=0}^{n} (-1)^k \binom{n}{k} = 0
\]](//latex.artofproblemsolving.com/6/b/3/6b354dd1e631011ca7b9f16cd7d91c512dc9ca08.png)
- Pascal's Identity:
![\[
\binom{n}{k} = \binom{n - 1}{k} + \binom{n - 1}{k - 1}
\]](//latex.artofproblemsolving.com/1/4/f/14f5861cc19608715ad04dbaa65821c244b43c67.png)
![\[
\binom{n}{k} = \binom{n}{n - k}
\]](http://latex.artofproblemsolving.com/b/4/2/b42a06fe41d2bdf0fb4d3fbec8ac0c702dfa896a.png)
![\[
\sum_{k=0}^{n} \binom{n}{k} = 2^n
\]](http://latex.artofproblemsolving.com/f/1/b/f1b4bf6764470e55f4258a215a620efa9680917f.png)
![\[
\sum_{k=0}^{n} (-1)^k \binom{n}{k} = 0
\]](http://latex.artofproblemsolving.com/6/b/3/6b354dd1e631011ca7b9f16cd7d91c512dc9ca08.png)
![\[
\binom{n}{k} = \binom{n - 1}{k} + \binom{n - 1}{k - 1}
\]](http://latex.artofproblemsolving.com/1/4/f/14f5861cc19608715ad04dbaa65821c244b43c67.png)
5. Generalized Binomial Theorem
When
is any real or complex number, the Binomial Theorem can be extended to the form of an infinite series:![\[
(1 + x)^\alpha = \sum_{k=0}^{\infty} \binom{\alpha}{k} x^k, \quad |x| < 1
\]](http://latex.artofproblemsolving.com/f/a/e/faeec2c23ce7ae02e584d322b4f5c325b7c9bdb0.png)
Where:
![\[
\binom{\alpha}{k} = \frac{\alpha (\alpha - 1) (\alpha - 2) \dots (\alpha - k + 1)}{k!}
\]](http://latex.artofproblemsolving.com/6/a/d/6ad28cdf508e9f6ef28e08240af15efc778a810e.png)
6. Applications of the Binomial Theorem
The Binomial Theorem has numerous applications:
- Combinatorics: Counting combinations and subsets.
- Algebra: Polynomial expansions and simplifications.
- Probability: Analyzing binomial distributions.
- Number Theory: Studying congruences and identities.
- Calculus: Approximating functions using Taylor series.
7. Example Problems
- Example 1: Find the coefficient of
in 
.
Using the Binomial Theorem:
![\[
\text{Coefficient of } x^5 = \binom{10}{5} = 252
\]](//latex.artofproblemsolving.com/8/6/0/86056127712ca26a9237bbf088cd2c8202b8c210.png)
- Example 2: Expand
.
![\[
(2x - 3y)^3 = \sum_{k=0}^{3} \binom{3}{k} (2x)^{3-k} (-3y)^k = 8x^3 - 36x^2y + 54xy^2 - 27y^3
\]](//latex.artofproblemsolving.com/6/8/7/6870080a6a51b599623b13158b7fc58e72759bc3.png)
in 
.![\[
\text{Coefficient of } x^5 = \binom{10}{5} = 252
\]](http://latex.artofproblemsolving.com/8/6/0/86056127712ca26a9237bbf088cd2c8202b8c210.png)
.![\[
(2x - 3y)^3 = \sum_{k=0}^{3} \binom{3}{k} (2x)^{3-k} (-3y)^k = 8x^3 - 36x^2y + 54xy^2 - 27y^3
\]](http://latex.artofproblemsolving.com/6/8/7/6870080a6a51b599623b13158b7fc58e72759bc3.png)
8. Conclusion
The Binomial Theorem is a powerful and elegant mathematical tool that connects algebra, combinatorics, and analysis. Its applications span various fields, making it one of the most important results in elementary and advanced mathematics.
9. References
- Wikipedia: Binomial Theorem
- Art of Problem Solving: AoPS Wiki: Binomial Theorem
- Concrete Mathematics, Knuth et al.
March 2025