Flash Cards: Binomial Theorem

The Binomial Theorem states that (a + b)ⁿ = ∑(from k=0 to n) nCk * a^(n-k) * b^k, where n is a non-negative integer.

Pascal's Triangle is an arrangement of binomial coefficients in a triangular format, where each number is the sum of the two directly above it.

There are (n + 1) terms in the expansion of (a + b)ⁿ, meaning one more than the index n.

In each term of the expansion, the power of 'a' decreases by 1, while the power of 'b' increases by 1.

(a + b)² = a² + 2ab + b².

Mathematical induction is used to prove the validity of the Binomial Theorem for all positive integers n.

(x + 2)³ = 3C0 * x³ + 3C1 * x²(2) + 3C2 * x(2)² + 3C3(2)³ = x³ + 12x² + 24x + 8.

(a + b)⁰ = 1 if a + b ≠ 0.

nCr = n! / [r!(n - r)!], where 0 ≤ r ≤ n.

(x - y)ⁿ = ∑(from k=0 to n) nCk * x^(n-k) * (-y)^k.

The sum of coefficients in (1 + x)ⁿ equals 2ⁿ, i.e., the value of (1 + 1)ⁿ.

A common mistake is not applying the sign changes properly when expanding (a - b)ⁿ.

(1 + x)ⁿ = ∑(from k=0 to n) nCk * x^k; (1 - x)ⁿ = ∑(from k=0 to n) (-1)ⁿCk * x^k.

The second term is given by nC1 * a^(n-1) * b.

The Binomial Theorem can be adapted for negative integers using its general form but requires additional considerations.

The first binomial coefficient for any n is nC0, which equals 1.

The fourth term is 4C3 * x^1 * y^3 = 4xy³.

The Binomial Theorem can be used in probability theory, such as calculating the likelihood of certain outcomes.