6.3 The Binomial Theorem

The Binomial Theorem is a powerful tool for expanding expressions like (a + b)^n. It's super useful in math and stats, helping us quickly find coefficients and terms without doing tons of multiplication.

This theorem connects to combinatorics by using Pascal's Triangle and binomial coefficients. It's also key in probability, especially for binomial distributions. Understanding this makes tackling complex expansions and probability problems way easier.

Expanding binomial expressions

The Binomial Theorem formula

Top images from around the web for The Binomial Theorem formula

• 

  Use the Binomial Theorem | College Algebra View original

  Is this image relevant?

• 

  Math Gems View original

  Is this image relevant?

• 

  Using the Binomial Theorem | College Algebra View original

  Is this image relevant?

• 

  Use the Binomial Theorem | College Algebra View original

  Is this image relevant?

• 

  Math Gems View original

  Is this image relevant?

1 of 3

Top images from around the web for The Binomial Theorem formula

• 

  Use the Binomial Theorem | College Algebra View original

  Is this image relevant?

• 

  Math Gems View original

  Is this image relevant?

• 

  Using the Binomial Theorem | College Algebra View original

  Is this image relevant?

• 

  Use the Binomial Theorem | College Algebra View original

  Is this image relevant?

• 

  Math Gems View original

  Is this image relevant?

1 of 3

• The Binomial Theorem states that for any real numbers $a$ and $b$ and any non-negative integer $n$, the expansion of $(a + b)^n$ is the sum of the terms $(n \text{ choose } k) * a^{n-k} * b^k$ for $k = 0, 1, 2, ..., n$
• The general form of the Binomial Theorem is: $(a + b)^n = \sum_{k=0}^n (n \text{ choose } k) * a^{n-k} * b^k$, where $(n \text{ choose } k)$ represents the binomial coefficient
• The Binomial Theorem provides a formula for expanding binomial expressions raised to any non-negative integer power without directly multiplying the binomial factors ($(x + y)^5$, [(2a - 3b)^4](https://www.fiveableKeyTerm:(2a_-_3b)^4))

Properties of the expanded binomial expression

• The number of terms in the expanded binomial expression is equal to $n + 1$, where $n$ is the exponent of the binomial
• The powers of $a$ in the expanded expression decrease from $n$ to $0$, while the powers of $b$ increase from $0$ to $n$, with the sum of the exponents in each term always equaling $n$
• The coefficients of the terms in the expanded expression are symmetric, meaning that the coefficients of the terms equidistant from the ends are equal
  • For example, in the expansion of $(a + b)^4$, the coefficients are 1, 4, 6, 4, 1

Coefficients in binomial expansions

Using Pascal's Triangle to determine coefficients

• Pascal's Triangle is a triangular array of numbers in which each number is the sum of the two numbers directly above it
• The rows of Pascal's Triangle are numbered starting from 0, and the entries in each row correspond to the binomial coefficients $(n \text{ choose } k)$ for a given value of $n$
• The $k$-th entry in the $n$-th row of Pascal's Triangle represents the coefficient of the term containing $a^{n-k} * b^k$ in the expansion of $(a + b)^n$
• The coefficients in the expanded binomial expression can be found by selecting the appropriate row of Pascal's Triangle based on the exponent $n$ and reading the entries from left to right (4th row for $(a + b)^4$: 1, 4, 6, 4, 1)

Properties of Pascal's Triangle

• The entries in the first and last columns are always 1
• The triangle is symmetric about its vertical center line
• The sum of the entries in each row is a power of 2, specifically $2^n$, where $n$ is the row number
  • For example, the sum of the entries in the 4th row (1, 4, 6, 4, 1) is $2^4 = 16$
• Pascal's Triangle provides a convenient way to determine the coefficients of terms in a binomial expansion without directly calculating the binomial coefficients using the formula

Applications of the Binomial Theorem

Binomial distributions and probability

• The Binomial Theorem is used to calculate probabilities in binomial distributions, which model the number of successes in a fixed number of independent trials with two possible outcomes (success or failure)
• In a binomial distribution, the probability of exactly $k$ successes in $n$ trials, denoted as $P(X = k)$, is given by the formula: $P(X = k) = (n \text{ choose } k) * p^k * (1 - p)^{n-k}$, where $p$ is the probability of success in a single trial
• The binomial coefficient $(n \text{ choose } k)$ in the probability formula represents the number of ways to choose $k$ successes from $n$ trials and can be calculated using the Binomial Theorem
• The expected value (mean) of a binomial distribution is $\mu = n * p$, and the variance is $\sigma^2 = n * p * (1 - p)$

Cumulative probability and generating functions

• The cumulative probability of a binomial distribution, $P(X \leq k)$, can be calculated by summing the individual probabilities for values of $X$ from $0$ to $k$
  • For example, to find $P(X \leq 2)$ in a binomial distribution with $n = 5$ and $p = 0.4$, calculate $P(X = 0) + P(X = 1) + P(X = 2)$
• The Binomial Theorem is also used to derive the moment-generating function and probability-generating function of a binomial distribution, which are useful for studying its properties and moments
  • The moment-generating function of a binomial distribution is $M_X(t) = (pe^t + 1 - p)^n$
  • The probability-generating function of a binomial distribution is $G_X(s) = (ps + 1 - p)^n$

Proof of the Binomial Theorem

Combinatorial argument

• The Binomial Theorem can be proved using combinatorial arguments by considering the number of ways to choose $k$ items from a total of $n$ items
• The proof relies on the idea that the coefficient of $a^{n-k} * b^k$ in the expansion of $(a + b)^n$ is equal to the number of ways to choose $k$ items from $n$ items, denoted as $(n \text{ choose } k)$ or $nC_k$
• The binomial coefficient $(n \text{ choose } k)$ can be expressed as $n! / (k! * (n-k)!)$, where $n!$ represents the factorial of $n$

Proof using sequences

• To prove the Binomial Theorem, consider the process of selecting $n$ items from a set containing two types of objects, $a$ and $b$, with repetition allowed
• Each selection can be represented as a sequence of $n$ choices, where each choice is either $a$ or $b$. The total number of such sequences is $2^n$, as there are two possible choices for each of the $n$ positions
• The number of sequences containing exactly $k$ occurrences of $b$ (and consequently, $n-k$ occurrences of $a$) is equal to $(n \text{ choose } k)$, as there are $(n \text{ choose } k)$ ways to choose the positions for the $k$ occurrences of $b$ among the $n$ positions
• The term $a^{n-k} * b^k$ represents all sequences with exactly $k$ occurrences of $b$, and the coefficient $(n \text{ choose } k)$ counts the number of such sequences
• Summing the terms $(n \text{ choose } k) * a^{n-k} * b^k$ for $k = 0, 1, 2, ..., n$ accounts for all possible sequences of length $n$ containing $a$ and $b$, which is equal to $(a + b)^n$
• Therefore, the Binomial Theorem, $(a + b)^n = \sum_{k=0}^n (n \text{ choose } k) * a^{n-k} * b^k$, is proved using combinatorial arguments