Binomial Theorem

Statement

The binomial theorem asserts that for any positive integer $n$, the expression $( a + b )^{n}$ can be expanded as a finite sum of $n + 1$ terms. Each term consists of a binomial coefficient multiplied by a power of $a$ and a power of $b$: \(( a + b )^{n} = ( \frac{n}{0} ) a^{n} b^{0} + ( \frac{n}{1} ) a^{n - 1} b^{1} + \ldots + ( \frac{n}{n - 1} ) a^{1} b^{n - 1} + ( \frac{n}{n} ) a^{0} b^{n}\)

The formula is composed of the following elements:

• $n$, the exponent term, is a positive integer $n \in \mathbb{N}^{+}$.
• $a$ is raised to a decreasing power, from $n$ to $0$.
• $b$ is raised to an increasing power from $0$ to $n$.
• The term $( \frac{n}{k} )$ is the binomial coefficient, where $k$ takes values between $0$ and $n$.

The coefficients $( \frac{n}{k} )$ appearing in the expansion correspond exactly to the entries of the $n$-th row of Pascal’s Triangle. The symmetry $( \frac{n}{k} ) = ( \frac{n}{n - k} )$ reflects the fact that choosing $k$ elements from $n$ is equivalent to leaving out $n - k$.

In its compact form, the binomial theorem can be expressed as the summation of $n + 1$ terms:

Binomial coefficient

The binomial coefficient represents the number of ways to choose $k$ items from a larger set of $n$ elements, ignoring the order of selection. In combinatorics, it is commonly referred to as n choose k, and is denoted by the notation:

• $n , k \in \mathbb{N}$.
• $n$ is the total number of elements in the set.
• $k$ is the number of items to be selected.
• $n !$ and $( n - k ) !$ are the factorials of the natural numbers $n$ and $n - k$ respectively.

Proof

There are two standard proofs of the theorem. The first is based on a combinatorial argument. The expansion of $( a + b )^{n}$ can be viewed as the product of $n$ identical factors:

Each term in the expanded product results from selecting either $a$ or $b$ from each factor. A term of the form $a^{n - k} b^{k}$ occurs when $b$ is chosen from exactly $k$ of the $n$ factors, and $a$ from the remaining $n - k$. The number of such selections is $( \frac{n}{k} )$, which enumerates the $k$-element subsets of the $n$ factors. Summing over all possible values of $k$ from $0$ to $n$ establishes the theorem.

The second proof uses mathematical induction on $n$. For $n = 1$, the identity becomes $( a + b )^{1} = a + b$, which is clearly valid. Assume the theorem holds for some integer $n \geq 1$. Multiplying both sides of the inductive hypothesis by $( a + b )$ yields:

Re-index the second sum by letting $j = k + 1$ and separate the edge terms as follows:

Apply Pascal’s identity, $( \frac{n}{k} ) + ( \frac{n}{k - 1} ) = ( \frac{n + 1}{k} )$, to the interior terms to obtain:

This expression is the binomial theorem for $n + 1$, which completes the induction.

Example 1

Expand $( x + 2 )^{4}$ using the binomial theorem, with $a = x$, $b = 2$, and $n = 4$.

Applying the formula:

Computing each term:

Summing all terms we obtain: \(( x + 2 )^{4} = x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16\)

Example 2

For a binomial involving subtraction, the binomial theorem applies with $b$ replaced by a negative value. The signs alternate in the expansion because $( - 1 )^{k}$ yields a positive value for even $k$ and a negative value for odd $k$. The expansion of $( x - 1 )^{5}$, where $a = x$, $b = - 1$, and $n = 5$, is as follows:

Each term in the expansion is computed as follows:

Summing all terms we have:

Selected resources

• University of Connecticut, W. Abikoff. The Binomial Theorem
• University of California S. Barbara, H. McGahagan. Induction and the Binomial Theorem
• MIT,D. Karger, N. Lynch. Binomial Coefficients and Identities
• Harvard University, C. Wang. Binomial Theorem and Combinatorial Proofs