Powers of Binomials: Pascal’s Triangle and the Binomial Theorem

Powers of Binomials

Binomial Expansion

The expansion of a binomial expression, such as $(a + b)^n$, follows a predictable pattern governed by the Binomial Theorem and Pascal’s Triangle. Understanding these concepts allows for efficient computation of terms and coefficients in binomial expansions.

• Number of Terms: The expansion of $(a + b)^n$ contains $n + 1$ terms.

• Exponent Patterns: The exponent of $a$ starts at $n$ and decreases by 1 in each successive term, while the exponent of $b$ starts at 0 and increases by 1.

• Sum of Exponents: In each term, the sum of the exponents of $a$ and $b$ is always $n$.

• Symmetry: The coefficients in the expansion are symmetric.

Pascal’s Triangle

Pascal’s Triangle is a triangular array of numbers where each row corresponds to the coefficients of the binomial expansion $(a + b)^n$. Each row begins and ends with 1, and each interior number is the sum of the two numbers directly above it in the previous row.

• Construction: Start with 1 at the top. Each subsequent row is formed by adding adjacent numbers from the previous row.

• Coefficients: The $n$th row gives the coefficients for $(a + b)^n$.

• Symmetry: Each row reads the same left to right as right to left (palindromic).

Example: Row for $(x + y)^7$

Row	Coefficients
7	1, 7, 21, 35, 35, 21, 7, 1

Expansion: $(x + y)^7 = x^7 + 7x^6y + 21x^5y^2 + 35x^4y^3 + 35x^3y^4 + 21x^2y^5 + 7xy^6 + y^7$

Combinations and Factorials

The coefficients in binomial expansions can be calculated using combinations, denoted $nCr$, which represent the number of ways to choose $r$ objects from $n$ distinct objects. Factorials are used in these calculations.

• Combination Formula: $nCr = \frac{n!}{r!(n - r)!}$

• Factorial: $n!$ is the product of all positive integers up to $n$. $0!$ is defined as 1.

Binomial Theorem

The Binomial Theorem provides a formula for expanding $(a + b)^n$ using combinations:

• General Formula:

$(a + b)^n = \sum_{r=0}^{n} nCr \cdot a^{n-r} b^r$

Expanded form:

$(a + b)^n = nC0 a^n b^0 + nC1 a^{n-1} b^1 + nC2 a^{n-2} b^2 + \ldots + nCn a^0 b^n$

Example: Expand $(w + \ell)^6$

$(w + \ell)^6 = 6C0 w^6 + 6C1 w^5 \ell + 6C2 w^4 \ell^2 + 6C3 w^3 \ell^3 + 6C4 w^2 \ell^4 + 6C5 w \ell^5 + 6C6 \ell^6$

Substitute values:

$= w^6 + 6w^5 \ell + 15w^4 \ell^2 + 20w^3 \ell^3 + 15w^2 \ell^4 + 6w \ell^5 + \ell^6$

Applications to Probability

Binomial expansions are used to calculate probabilities in situations where there are two possible outcomes (e.g., win/loss, heads/tails). If outcomes are equally likely, the coefficients represent the number of ways each outcome can occur.

• Probability Calculation: Divide the coefficient for the desired outcome by the total number of possible outcomes ($2^n$ for $n$ trials).

• Example: Probability of 3 wins and 3 losses in 6 games: $\frac{20}{64} \approx 31\%$

Expanding Binomials with Coefficients Other Than 1

When the binomial has coefficients other than 1, the expansion involves multiplying the terms by these coefficients, and the resulting coefficients may not be symmetric.

Example: Expand $(2c - 6d)^4$

$(2c - 6d)^4 = 4C0 (2c)^4 + 4C1 (2c)^3 (-6d) + 4C2 (2c)^2 (-6d)^2 + 4C3 (2c) (-6d)^3 + 4C4 (-6d)^4$

Calculate each term:

• $4C0 (2c)^4 = 1 \cdot 16c^4 = 16c^4$

• $4C1 (2c)^3 (-6d) = 4 \cdot 8c^3 \cdot (-6d) = -192c^3d$

• $4C2 (2c)^2 (-6d)^2 = 6 \cdot 4c^2 \cdot 36d^2 = 864c^2d^2$

• $4C3 (2c) (-6d)^3 = 4 \cdot 2c \cdot (-216d^3) = -1728cd^3$

• $4C4 (-6d)^4 = 1 \cdot 1296d^4 = 1296d^4$

Final expansion:

$(2c - 6d)^4 = 16c^4 - 192c^3d + 864c^2d^2 - 1728cd^3 + 1296d^4$

Practice Problems and Applications

• Expand binomials using Pascal’s Triangle and the Binomial Theorem.

• Apply binomial expansions to probability scenarios (e.g., games, coin tosses, selection problems).

• Find specific terms in binomial expansions (e.g., term where exponent of $a$ is 5 in $(a + b)^{12}$).

• Approximate values using binomial expansion (e.g., $(1 + 0.02)^{10}$ to estimate $(1.02)^{10}$).

Summary Table: Binomial Expansion Methods

Method	Description	When to Use
Pascal’s Triangle	Use rows to find coefficients for small $n$	Quick expansions, small exponents
Binomial Theorem	Use combination formula for coefficients	Any exponent, especially large $n$ or non-unit coefficients

Key Terms

• Binomial: An algebraic expression with two terms, e.g., $(a + b)$.

• Binomial Expansion: The process of expressing $(a + b)^n$ as a sum of terms.

• Pascal’s Triangle: A triangular array of numbers used to find binomial coefficients.

• Combination ($nCr$): The number of ways to choose $r$ objects from $n$.

• Factorial ($n!$): The product of all positive integers up to $n$.

Additional info:

• For probability problems with unequal likelihoods, substitute the actual probabilities for $a$ and $b$ in the binomial expansion.

• Pascal’s Triangle is palindromic because the combination formula $nCr = nC(n - r)$, making the coefficients symmetric.

• When expanding binomials with coefficients other than 1, use the Binomial Theorem for accuracy.