Binomial Theorem: Set 1

Learning Objectives Define and calculate factorials and combinations (nCk). Generate rows of Pascal's Triangle and explain its relationship to binomial coefficients. State the Binomial Theorem for positive integer exponents. Apply the Binomial Theorem to expand binomials of the form (a + b)^n. Correctly expand binomials involving subtraction and numerical coefficients, such as (cx - dy)^n. Use the general term formula to find a specific term in a binomial expansion. Ever tried to calculate (x + y)^7 by hand? 🤯 There's a powerful shortcut that reveals a beautiful mathematical pattern! This lesson introduces the Binomial Theorem, a formula that allows you to quickly expand binomials raised to any positive integer power. You'll discover its connection to combin...

TermDefinitionExample BinomialA polynomial with exactly two terms.x + 5, or 2a - 3b FactorialThe product of all positive integers up to a given non-negative integer 'n', denoted as n!. By definition, 0! = 1.5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 Combination (Binomial Coefficient)The number of ways to choose 'k' items from a set of 'n' items without regard to order. It is denoted by \binom{n}{k} or nCk.\binom{5}{2} represents the number of ways to choose 2 items from a set of 5, which is 10. Pascal's TriangleA triangular array of numbers where each number is the sum of the two numbers directly above it. The n-th row (starting from row 0) provides the binomial coefficients for the expansion of (a+b)^n.The 4th row is 1, 4, 6, 4, 1, which are the c...

Combination Formula \binom{n}{k} = \frac{n!}{k!(n-k)!} Use this formula to calculate the binomial coefficient, which represents the coefficient of a specific term in the expansion. 'n' is the power of the binomial, and 'k' is the term index (starting from 0). The Binomial Theorem (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k This is the general formula for expanding any binomial (a+b) raised to the power n. It is a sum of terms, where for each term, the power of 'a' decreases from n to 0, and the power of 'b' increases from 0 to n. General Term Formula T_{k+1} = \binom{n}{k} a^{n-k} b^k Use this formula to find a specific term in the expansion without calculating the entire series. Note that this formula gives the (k+1)-th...