X x n x
(The same generalization also applies to complex exponents.) In this generalization, the finite sum is replaced by an infinite series.In order to do this, one needs to give meaning to binomial coefficients with an arbitrary upper index, which cannot be done using the usual formula with factorials.
Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting k objects out of n without replacement, were of interest to ancient Indian mathematicians.
The earliest known reference to this combinatorial problem is the Chandaḥśāstra by the Indian lyricist Pingala (c. Blaise Pascal studied the eponymous triangle comprehensively in the treatise Traité du triangle arithmétique (1653).
However, the pattern of numbers was already known to the European mathematicians of the late Renaissance, including Stifel, Niccolò Fontana Tartaglia, and Simon Stevin.
In this form, the formula reads The binomial coefficients 1, 2, 1 appearing in this expansion correspond to the second row of Pascal's triangle.
(The top "1" of the triangle is considered to be row 0, by convention.) The coefficients of higher powers of Around 1665, Isaac Newton generalized the binomial theorem to allow real exponents other than nonnegative integers.