12.4 Binomial Theorem

Pascal's Triangle and the Binomial Theorem give you efficient ways to expand expressions like $(a+b)^n$ without multiplying everything out by hand. These tools connect algebra to combinatorics and show up again in probability, so getting comfortable with them now pays off.

Pascal's Triangle and the Binomial Theorem

Construction of Pascal's Triangle

Pascal's Triangle is a triangular array of numbers where each entry is the sum of the two numbers directly above it. The first and last numbers in every row are always 1.

Here are the first several rows:

• Row 0: 1
• Row 1: 1, 1
• Row 2: 1, 2, 1
• Row 3: 1, 3, 3, 1
• Row 4: 1, 4, 6, 4, 1

Each row gives you the coefficients for expanding $(a+b)^n$. Row $n$ contains the coefficients of $(a+b)^n$. Within that row, the powers of $a$ decrease from $n$ down to 0, while the powers of $b$ increase from 0 up to $n$.

To expand a binomial using Pascal's Triangle:

1. Find the row that matches your exponent $n$
2. Write each term using the row's numbers as coefficients
3. Attach the correct powers of $a$ and $b$ to each coefficient ($a$ starts at power $n$ and decreases; $b$ starts at power 0 and increases)

Example: Expand $(x+2)^4$ using Row 4: (1, 4, 6, 4, 1).

$= 1 \cdot x^4 \cdot 2^0 + 4 \cdot x^3 \cdot 2^1 + 6 \cdot x^2 \cdot 2^2 + 4 \cdot x^1 \cdot 2^3 + 1 \cdot x^0 \cdot 2^4$

$= x^4 + 8x^3 + 24x^2 + 32x + 16$

Methods for Binomial Coefficients

Binomial coefficients are the numbers that appear in Pascal's Triangle. They're written as $\binom{n}{k}$ (read "n choose k"), where $n$ is the row and $k$ is the position within that row, both counting from 0.

You can find binomial coefficients three ways:

• Pascal's Triangle: Look up the number at row $n$, position $k$
• Factorial formula: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, where $n!$ means $n \times (n-1) \times \cdots \times 1$
• Recursive relation: $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$ (this is just the "add the two numbers above" rule stated as a formula)

Two useful properties to know:

• Symmetry: $\binom{n}{k} = \binom{n}{n-k}$. For example, $\binom{7}{2} = \binom{7}{5}$. This is why Pascal's Triangle looks the same on both sides.
• Row sum: All the numbers in row $n$ add up to $2^n$. Row 4 sums to $1+4+6+4+1 = 16 = 2^4$.

Examples:

• Using the formula: $\binom{6}{3} = \frac{6!}{3! \cdot 3!} = \frac{720}{6 \cdot 6} = \frac{720}{36} = 20$
• Using the recursive relation: $\binom{7}{4} = \binom{6}{3} + \binom{6}{4} = 20 + 15 = 35$

Applications of the Binomial Theorem

The Binomial Theorem is the general formula for expanding $(a+b)^n$:

$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} \, a^{n-k} \, b^k$

The summation symbol $\sum$ tells you to add up all the terms from $k=0$ through $k=n$. Each term has three parts: a binomial coefficient, a power of $a$, and a power of $b$.

To expand a binomial using this theorem:

1. Identify $a$, $b$, and $n$. Be careful with signs: if you have $(2x - 3)^3$, then $b = -3$, not 3.

2. Calculate the binomial coefficients $\binom{n}{0}, \binom{n}{1}, \ldots, \binom{n}{n}$

3. Build each term: multiply the coefficient by $a^{n-k}$ and $b^k$

4. Simplify by evaluating the powers and combining

Example 1: Expand $(2x-3)^3$.

Here $a = 2x$, $b = -3$, $n = 3$. The coefficients from Row 3 are 1, 3, 3, 1.

$= 1(2x)^3(-3)^0 + 3(2x)^2(-3)^1 + 3(2x)^1(-3)^2 + 1(2x)^0(-3)^3$

$= 8x^3 - 36x^2 + 54x - 27$

Notice how the signs alternate because $b$ is negative. Odd powers of $-3$ are negative; even powers are positive.

Example 2: Find the coefficient of $x^5$ in $(3x+2)^7$.

You don't need to expand the whole thing. The term with $x^5$ has $a^5 = (3x)^5$, which means $n-k = 5$, so $k = 2$.

$\binom{7}{2}(3x)^5(2)^2 = 21 \cdot 243x^5 \cdot 4 = 20{,}412 \, x^5$

The coefficient is 20,412.

Additional Concepts and Applications

• Combinatorics: The binomial coefficient $\binom{n}{k}$ counts the number of ways to choose $k$ items from $n$ items. This counting interpretation is why the same numbers appear in both Pascal's Triangle and the Binomial Theorem.
• Polynomial expansion: The Binomial Theorem is a specific, efficient method for expanding $(a+b)^n$. For higher powers like $n = 10$, it saves enormous amounts of work compared to repeated multiplication.
• Algebraic identity: The theorem itself is an identity, meaning it holds true for all values of $a$, $b$, and non-negative integers $n$.
• Binomial distribution: In probability, when an experiment has two outcomes (success/failure) repeated $n$ times, the binomial coefficients determine the probabilities. You'll see this connection in statistics courses.