- $x+y$

- $2a-3b$

- $x^2+y^2$

- $5p+q^3$

- For $n > 1$: $n! = n \times (n-1) \times (n-2) \times \dots \times 3 \times 2 \times 1$

- $0! = 1$ (by definition, to maintain consistency in mathematical formulas)

- $1! = 1$

- $3! = 3 \times 2 \times 1 = 6$

- $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$

- Notation: $\binom{n}{k}$ or $C(n,k)$

- Formula: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$

- $\binom{n}{0} = 1$ (There is only one way to choose 0 items from $n$ items)

- $\binom{n}{n} = 1$ (There is only one way to choose $n$ items from $n$ items)

- $\binom{n}{k} = \binom{n}{n-k}$ (Symmetry property: choosing $k$ items is the same as choosing to leave $n-k$ items)

- $\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{(2 \times 1)(3 \times 2 \times 1)} = \frac{120}{2 \times 6} = \frac{120}{12} = 10$

- $\binom{4}{0} = \frac{4!}{0!(4-0)!} = \frac{4!}{1 \times 4!} = 1$

- The sum of the numbers in row $n$ is $2^n$.

- Each number in the triangle corresponds to a binomial coefficient $\binom{n}{k}$, where $n$ is the row number (starting from 0) and $k$ is the position in that row (starting from 0).

- Row 0 ($n=0$): 1 ($\binom{0}{0}$)

- Row 1 ($n=1$): 1 1 ($\binom{1}{0}$, $\binom{1}{1}$)

- Row 2 ($n=2$): 1 2 1 ($\binom{2}{0}$, $\binom{2}{1}$, $\binom{2}{2}$)

- Row 3 ($n=3$): 1 3 3 1 ($\binom{3}{0}$, $\binom{3}{1}$, $\binom{3}{2}$, $\binom{3}{3}$)

- Row 4 ($n=4$): 1 4 6 4 1 ($\binom{4}{0}$, $\binom{4}{1}$, $\binom{4}{2}$, $\binom{4}{3}$, $\binom{4}{4}$)

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

- Expanded form: $(a+b)^n = \binom{n}{0}a^n b^0 + \binom{n}{1}a^{n-1}b^1 + \binom{n}{2}a^{n-2}b^2 + \dots + \binom{n}{n-1}a^1 b^{n-1} + \binom{n}{n}a^0 b^n$

- The number of terms in the expansion of $(a+b)^n$ is $n+1$.

- The sum of the powers of $a$ and $b$ in each term is always $n$.

- Expand $(x+y)^3$:

$(x+y)^3 = \binom{3}{0}x^3y^0 + \binom{3}{1}x^2y^1 + \binom{3}{2}x^1y^2 + \binom{3}{3}x^0y^3$

$= 1 \cdot x^3 \cdot 1 + 3 \cdot x^2 \cdot y + 3 \cdot x \cdot y^2 + 1 \cdot 1 \cdot y^3$

$= x^3 + 3x^2y + 3xy^2 + y^3$

- Expand $(2x-3y)^4$:

$(2x-3y)^4 = \binom{4}{0}(2x)^4(-3y)^0 + \binom{4}{1}(2x)^3(-3y)^1 + \binom{4}{2}(2x)^2(-3y)^2 + \binom{4}{3}(2x)^1(-3y)^3 + \binom{4}{4}(2x)^0(-3y)^4$

$= 1(16x^4)(1) + 4(8x^3)(-3y) + 6(4x^2)(9y^2) + 4(2x)(-27y^3) + 1(1)(81y^4)$

$= 16x^4 - 96x^3y + 216x^2y^2 - 216xy^3 + 81y^4$

- The $(r+1)^{th}$ term, denoted as $T_{r+1}$, is given by: $T_{r+1} = \binom{n}{r} a^{n-r} b^r$

- Find the 3rd term of $(x+2y)^5$.

Here, $n=5$, $a=x$, $b=2y$. For the 3rd term, $r+1=3$, so $r=2$.

$T_3 = \binom{5}{2} x^{5-2} (2y)^2$

$T_3 = 10 \cdot x^3 \cdot (4y^2)$

$T_3 = 40x^3y^2$

- Approximations: It can be used to approximate values of expressions like $(1.02)^4$ by writing them as $(1+0.02)^4$ and expanding.

- Algebraic Identities: It can be used to prove various algebraic identities by setting specific values for $a$ and $b$. For example, setting $a=1, b=1$ in $(a+b)^n$ shows that $\sum_{k=0}^{n} \binom{n}{k} = 2^n$.

• Binomial Expression: An algebraic expression consisting of exactly two terms, typically joined by addition or subtraction, e.g., $x+y$ or $2a-3b$.

• Factorial ($n!$): The product of all positive integers less than or equal to a given non-negative integer $n$. Defined as $n! = n \times (n-1) \times \dots \times 2 \times 1$ for $n>1$, with $0! = 1$ and $1! = 1$.

• Binomial Coefficient ($\binom{n}{k}$ or $C(n,k)$): The coefficient of a term in the binomial expansion of $(a+b)^n$. It represents the number of ways to choose $k$ items from a set of $n$ items without regard to order, calculated as $\frac{n!}{k!(n-k)!}$.

• Binomial Theorem: A mathematical theorem that provides a formula for expanding binomial expressions of the form $(a+b)^n$ for any positive integer $n$. The formula is $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$.

• General Term ($T_{r+1}$): The formula for finding any specific term in a binomial expansion without performing the full expansion. The $(r+1)^{th}$ term is given by $T_{r+1} = \binom{n}{r} a^{n-r} b^r$.

• Index/Power ($n$): The exponent to which the binomial expression is raised, indicating the degree of the expansion.

- Problem: Expand $(2x+y)^3$.

- Explanation: Using the Binomial Theorem $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$ with $a=2x$, $b=y$, and $n=3$.

$(2x+y)^3 = \binom{3}{0}(2x)^3(y)^0 + \binom{3}{1}(2x)^2(y)^1 + \binom{3}{2}(2x)^1(y)^2 + \binom{3}{3}(2x)^0(y)^3$

$= 1(8x^3)(1) + 3(4x^2)(y) + 3(2x)(y^2) + 1(1)(y^3)$

$= 8x^3 + 12x^2y + 6xy^2 + y^3$

- This example demonstrates the direct application of the theorem to expand a binomial, showing how the powers of $a$ decrease and $b$ increase, and how binomial coefficients are used.

- Problem: Find the 4th term in the expansion of $(3x-2y)^6$.

- Explanation: For the 4th term, we use the general term formula $T_{r+1} = \binom{n}{r} a^{n-r} b^r$. Here, $n=6$, $a=3x$, $b=-2y$. Since we want the 4th term, $r+1=4$, so $r=3$.

$T_4 = \binom{6}{3} (3x)^{6-3} (-2y)^3$

$T_4 = \frac{6!}{3!3!} (3x)^3 (-2y)^3$

$T_4 = 20 (27x^3) (-8y^3)$

$T_4 = 20 \times 27 \times (-8) x^3y^3$

$T_4 = -4320x^3y^3$

- Problem: Use the Binomial Theorem to approximate $(1.01)^5$ to four decimal places.

- Explanation: We can rewrite $(1.01)^5$ as $(1+0.01)^5$. Using the Binomial Theorem for $(a+b)^n$ with $a=1$, $b=0.01$, and $n=5$:

$(1+0.01)^5 = \binom{5}{0}(1)^5(0.01)^0 + \binom{5}{1}(1)^4(0.01)^1 + \binom{5}{2}(1)^3(0.01)^2 + \binom{5}{3}(1)^2(0.01)^3 + \dots$

$= 1(1)(1) + 5(1)(0.01) + 10(1)(0.0001) + 10(1)(0.000001) + \dots$

$= 1 + 0.05 + 0.0010 + 0.000010 + \dots$

$1 + 0.05 + 0.0010 + 0.000010 = 1.051010$

Rounding to four decimal places, $(1.01)^5 \approx 1.0510$.

The document begins by defining a binomial expression as an algebraic expression with two terms, such as $x+y$ or $2a-3b$. It then introduces factorials, denoted by $n!$, as the product of all positive integers up to $n$, with $0! = 1$ and $1! = 1$. Factorials are presented as a fundamental building block for calculating binomial coefficients.

A significant portion of the document is dedicated to Pascal's Triangle, a visual and intuitive method for generating binomial coefficients. The document details its construction, where each number is the sum of the two numbers directly above it, and highlights its properties, including symmetry and the fact that the sum of elements in row $n$ is $2^n$. Pascal's Triangle serves as an excellent tool for understanding the pattern of coefficients for smaller values of $n$.

The core of the document is the Binomial Theorem itself, which provides a general formula for expanding $(a+b)^n$ for any positive integer $n$. The theorem is stated as $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$, and its expanded form is meticulously presented. Key observations about the expansion are emphasized: there are $n+1$ terms, the sum of the powers of $a$ and $b$ in each term is always $n$, and the coefficients follow the pattern of Pascal's Triangle. Detailed examples, such as expanding $(x+y)^3$ and $(2x-3y)^4$, are provided to illustrate the application of the theorem step-by-step.

Beyond full expansions, the document introduces the concept of the general term of a binomial expansion. The formula for the $(r+1)^{th}$ term, $T_{r+1} = \binom{n}{r} a^{n-r} b^r$, is given, enabling students to find any specific term in an expansion without needing to compute all preceding terms. An example demonstrates finding the 3rd term of $(x+2y)^5$.

Finally, the document explores various applications of the Binomial Theorem. These include its use in numerical approximations (e.g., approximating $(1.02)^4$), its fundamental role in probability theory (specifically the binomial distribution), its connection to combinatorics, and its utility in proving algebraic identities. The conclusion reiterates the theorem's power and versatility across different mathematical domains.