Binomial Theorem
Binomial Theorem Concepts to Remember
Pascal’s Triangle:
- A triangular arrangement of binomial coefficients.
- Each number in the triangle is the sum of the two numbers directly above it.
- nth row represents the coefficients of the (n+1)th term of the binomial expansion.
General term of binomial expansion:
- $$T_r = ^nC_r X^n-rY^r$$
- Where
^nC_ris the binomial coefficient. - (n)is the exponent of the binomial raised to the power of
n-r. -(Y)is the second term of the binomial raised to the power of (r). -(r) represents the specific term in the expansion
Binomial coefficients:
- The numbers that appear in Pascal’s Triangle.
- Represent the coefficient of each term in the binomial expansion.
- Calculated using the formula: ^^nCr = n!/r!(n - r)!.
Properties of binomial coefficients:
- Symmetric property: ^nCr = ^nCn-r. -Pascal’s rule: ^nCr = ^n-1Cr + ^n-1Cr-1.
Applications of binomial theorem:
- Approximating probabilities in probability distributions (normal, binomial, Poisson).
- Expanding algebraic expressions.
- Calculating combinations and permutations.
- Simplifying complex expressions involving powers and products of binomials.
