BINOMIAL THEOREM

statement that, for any positive integer n, the nth power of the sum of two numbers a and b may be expressed as the sum of n + 1 terms of the form in the sequence of terms, the index r takes on the successive values 0, 1, 2, . . . , n. The coefficients, called the binomial coefficients, are defined by the formula in which n! (called n factorial) is the product of the first n natural numbers 1, 2, 3, . . . n (and where 0! is defined as equal to 1). The coefficients may also be found in the array often called Pascal's triangle by finding the rth entry of the nth row (counting begins with a zero in both directions). Each entry in the interior of Pascal's triangle is the sum of the two entries above it. Isaac Newton stated in 1676, without proof, the general form of the theorem (for any real number n), although mathematicians had been aware of simple cases long before his time. A proof by Jakob Bernoulli was published in 1713, after Bernoulli's death.