The binomial expansion is the process of expanding expressions of the form $(a + b)^n$, where $n$ is a non-negative integer. The formula for the expansion is given by the Binomial Theorem:

Where:

• $\binom{n}{k}$ is the binomial coefficient, read as "n choose k," and is given by:

• $n!$ is the factorial of $n$.
• $a^{n-k}$ is the term involving $a$, and $b^k$ is the term involving $b$.

The binomial expansion of $(a + b)^n$ will look like this:

Each term in the expansion consists of:

1. Binomial Coefficient: $\binom{n}{k}$
2. A power of $a$, starting from $a^n$ and decreasing to $a^0$.
3. A power of $b$, starting from $b^0$ and increasing to $b^n$.

4. Example 1: Expand $(x + y)^2$ :

Using the Binomial Theorem:

• The first term is $x^2$, with a binomial coefficient of $\binom{2}{0} = :success[1]$ .
• The second term is $2xy$, with a binomial coefficient of $\binom{2}{1} = :success[2]$ .
• The third term is $y^2$, with a binomial coefficient of $\binom{2}{2} = :success[1]$ .

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

Using the Binomial Theorem:

• The first term is $x^3$, with a binomial coefficient of $\binom{3}{0} = :success[1]$.
• The second term is $3x^2$, with a binomial coefficient of $\binom{3}{1} = :success[3]$.
• The third term is $3x$, with a binomial coefficient of $\binom{3}{2} = :success[3]$.
• The fourth term is $1$, with a binomial coefficient of $\binom{3}{3} = :success[1]$.

Binomial Coefficients:

The binomial coefficients $\binom{n}{k}$ are the numbers that appear in Pascal's Triangle, and they can be computed using the formula:

Where $n!$ (factorial of $n$) is the product of all positive integers up to $n$.

6. The total number of terms in the expansion of $(a + b)^n$ is n + 1 .
7. The sum of the exponents of $a$ and $b$ in each term is always $n$.
8. The binomial expansion can be applied to both positive and negative terms as long as $n$ is a non-negative integer.

9. For $(a - b)^n$: The expansion follows the same pattern as $(a + b)^n$, except the signs alternate.

10. For fractional or negative exponents, the binomial expansion becomes an infinite series and requires more advanced methods (not covered in basic binomial expansion).

The binomial coefficient (combinations) is given by: ${}^nC_r = {}_nC_r = \binom{n}{r} = \frac{n!}{r!(n-r)!}$

$\dbinom{n}{r} = \dfrac{n!}{r!(n-r)!}$

Example Calculations

1. Choosing $2$ out of $4$: $\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3 \times 2 \times 1}{2 \times 1 \times 2 \times 1} = \frac{24}{4} = :success[6]$

Table of Combinations

The formula for combinations ensures that we count the number of ways to choose items without regard to the order of selection.