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1. Factorial \(n!\):

For \(n = 1, 2, 3, \dots\), factorial \(n\) is defined as:

\[

n! = n \cdot (n-1) \cdot (n-2) \cdot \dots \cdot 2 \cdot 1

\]

Zero factorial is defined as:

\[

0! = 1

\]

\[

4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24

\]

\[

5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120

\]

\[

6! = 6 \cdot 5! = 720

\]

2. Binomial Expansion:

For \(n = 1, 2, 3, \dots\), the binomial expansion is:

\[

(x + y)^n = \binom{n}{0}x^n + \binom{n}{1}x^{n-1}y + \binom{n}{2}x^{n-2}y^2 + \dots + \binom{n}{n}y^n

\]

where \(\binom{n}{k}\) is the binomial coefficient.

Example:

\[

(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4

\]

3. Binomial Coefficient Definition:

The binomial coefficients are defined as:

\[

\binom{n}{k} = \frac{n!}{k!(n-k)!}, \quad k = 0, 1, 2, \dots, n

\]

Example:

\[

\binom{9}{4} = \frac{9 \cdot 8 \cdot 7 \cdot 6}{1 \cdot 2 \cdot 3 \cdot 4} = 126

\]

4. Pascal’s Triangle:

The binomial coefficients can be arranged in Pascal’s triangle, where each number is the sum of the two numbers directly above it. For example:

\[

10 = 4 + 6, \quad 15 = 5 + 10, \quad 20 = 10 + 10

\]

5. Basic Property:

\[

\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}

\]

6. Summation Property:

\[

\sum_{k=0}^{n} \binom{n}{k} = 2^n

\]

7. Alternating Sum Property:

\[

\sum_{k=0}^{n} (-1)^k \binom{n}{k} = 0

\]

8. Symmetry Property:

\[

\binom{n}{k} = \binom{n}{n-k}

\]

The binomial formula is a method to expand expressions of the form \((a + b)^n\), where \(n\) is a non-negative integer. This expansion is given by the Binomial Theorem, which states:

\[

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

\]

Where:

– \(n\) is the power to which the binomial is raised.

– \(k\) is the index of summation, running from 0 to \(n\).

– \(\binom{n}{k}\) is the binomial coefficient, which represents the number of ways to choose \(k\) elements from a set of \(n\) elements.

Binomial Coefficients:

The binomial coefficient \(\binom{n}{k}\) is calculated using the formula:

\[

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

\]

Where:

– \(n!\) (n factorial) is the product of all positive integers up to \(n\).

– \(k!\) is the factorial of \(k\).

– \((n-k)!\) is the factorial of \(n-k\).

These coefficients are the same numbers that appear in Pascal’s Triangle, and they describe the number of ways to choose a subset of \(k\) elements from a larger set of \(n\) elements.

Example (Binomial Expansion):

Let’s expand \((a + b)^3\) using the binomial formula:

\[

(a + b)^3 = \binom{3}{0}a^3b^0 + \binom{3}{1}a^2b^1 + \binom{3}{2}a^1b^2 + \binom{3}{3}a^0b^3

\]

This simplifies to:

\[

(a + b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3

\]

Applications:

– Combinatorics: Binomial coefficients are used to calculate combinations in probability and counting problems.

– Algebra: The binomial formula helps simplify the expansion of powers of binomials.

– Calculus: The binomial expansion is useful for approximations, especially when using Taylor or Maclaurin series.

Understanding binomial coefficients and the binomial theorem is essential in algebra, probability, and various branches of mathematics.