Binomial Expansion Formulas
To find binomial coefficients, we can also use Pascal’s Triangle.
Binomial Expansion
Important points to remember
- The total number of terms in the expansion of (x+y)n is (n+1)
- The sum of exponents of x and y is always n.
- nC0, nC1, nC2, … .., nCn are called binomial coefficients and also represented by C0, C1, C2, ….., Cn
- The binomial coefficients, which are equidistant from the beginning and from the ending, are equal, i.e., nC0 = nCn, nC1 = nCn-1 , nC2 = nCn-2 ,….. etc.
Binomial Theorem for Any Index
Let n be a rational number and x be a real number, such that | x | < 1. Then,
If the rational number and -1 < x <1, then,
- (1 − x)-1 = 1 + x + x2 + x3 + … + xr + … ∞
- (1 + x)-1 = 1 – x + x2 – x3 + … (−1)r xr+ … ∞
- (1 − x)-2 = 1 + 2x + 3x2 − 4x3 + … + (r + 1)xr + … ∞
- (1 + x)-2 = 1 − 2x + 3x2 − 4x3 + … + (−1)r (r + 1)xr + … ∞
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