Binomial Theorem - Statement of The Theorem

Statement of The Theorem

According to the theorem, it is possible to expand any power of x + y into a sum of the form

(x+y)^n = {n \choose 0}x^n y^0 + {n \choose 1}x^{n-1}y^1 + {n \choose 2}x^{n-2}y^2 + \cdots + {n \choose n-1}x^1 y^{n-1} + {n \choose n}x^0 y^n,

where each is a specific positive integer known as binomial coefficient. This formula is also referred to as the Binomial Formula or the Binomial Identity. Using summation notation, it can be written as

(x+y)^n = \sum_{k=0}^n {n \choose k}x^{n-k}y^k = \sum_{k=0}^n {n \choose k}x^{k}y^{n-k}.

The final expression follows from the previous one by the symmetry of x and y in the first expression, and by comparison it follows that the sequence of binomial coefficients in the formula is symmetrical.

A variant of the binomial formula is obtained by substituting 1 for y, so that it involves only a single variable. In this form, the formula reads

or equivalently

Read more about this topic:  Binomial Theorem

Famous quotes containing the words statement of the, statement of, statement and/or theorem:

    “Eroticism has its own moral justification because it says that pleasure is enough for me; it is a statement of the individual’s sovereignty.”
    —Mario Vargas Llosa (b. 1936)

    “It is commonplace that a problem stated is well on its way to solution, for statement of the nature of a problem signifies that the underlying quality is being transformed into determinate distinctions of terms and relations or has become an object of articulate thought.”
    —John Dewey (1859–1952)

    “After the first powerful plain manifesto
    The black statement of pistons, without more fuss
    But gliding like a queen, she leaves the station.”
    —Stephen Spender (1909–1995)

    “To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
    —Albert Camus (1913–1960)